HYDRODYNAMICS
A STUDY IN LOGIC, FACT AND
SIMILITUDE
Revised edition
by
GARRETT BIRKHOFF
PRINCETON, NEWJERSEY
PRINCETON UNIVERSITY PRESS
1960

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or Kpaea1,1x ycnoeHii. Cw.Ha d 8 m 8 rd J.,
Lectures on Cauchy's problem
Yale Univ. Press, 1923, Clfl 32.
'

16
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Hccne.noeaHHIO HX npHeMneMOCTH.
1) }!JUI npOCTOTbl H3JIOlKeHHR Ml,I He KacaeMCR 98KOHa coxpaHeHHR 9ReprHH
e apyrHX TepMOAHHaMH'leCKHX coo6palKeHRA, KOTOpble TaK>Ke MO>KHO npH&.ne'lb
(CM. § 14).
2) 1JarpaH>K )I(. JI.,
AHa,1HTll'lf>':KBJf MeXBHIIKB, r. II, M. - n., 1950.
CTp. 307,

§ 2. TuiJpoiJuHaMu"ec ,cue napaiJo,ccbl
17
§ 2. f HAPOAHH8MHqeCKHe napuo1cw
Ha .Ilene B psrn.e cnyt1aes ypasHeHHSI 3Anepa 6blJIH npoHHTerpu-
posaHbl, HO pe3yJibTaTbl pacqeTOB pe3KO pacXOJlHJIHCb c ua6JIIOJleHH-
SIMH, l.lTO SIBHO npOTHBopellHT MHeHHIO JlarpaH>K&. B rHJlPOJJ.HHaMHKe
TaKHe HeCOMHeHHble nporueope"UJI Me>KJlY 3KCnepHMeHTaJlbHblMH
)laHHl>IMH H 3aKJIIOtieHHSIMH, OCHOBaHHblMH Ha npaBJJ.OilOJl06HblX
paccy>K)leHHHX, Ha3blBaIOTCSI napaao,ccaMU, H B )laJibHeAmeM 3TOT
TepMHH (SyJJ.eT ynoTpe6JIHTbCSI HMeHHO B TaKOM CMblCJie.
3TH napaJJ.OKCbl 6blJIH npe)lMeTOM MHOrHX OCTpOT. TaK, He)laBHO
6blJIO CKa3a.HO 1), l .lTO B ,lleBSITHaJJ.uaTOM seKe «rH.llPO.llHHaMHKH pa3-
JleJISIJIHCb Ha HH:1KeHepoB-rH.11paBJIHKOB, KOTOpble Ha6n10.11aJIH TO,
l.lTO HeJib3SI 6hlJIO o6'bHCHHTb, H MaTeMaTHKOB, KOTOpble o6'bHCHSIJIH
TO, l.lTO HeJlb3ff 6blJIO Ha6n10.11aTb>. (HaM Ka>KeTCSI, l .lTO npe.11cTaBH·
TMH o6oax BH.!10B see eme BCTpe11a10TCH.)
. Ila H CH.11HeH foJibJl·
lllTeHH 3aMeTHJI, '!TO BCIO KHHry JlaM6a [7] MO:IKHO npO'IHTaTb, He
npe.11cTaBJIHH ce6e, '!TO so.11a ..• MOKpasrl
Tenepb o6bl'tHO 33SIBJISIIOT, llTO nO.ll06Hble napa.llOKCbl B03HHKaJOT
H3·3a OTJIH'IHSI peaJibHblX :IKHJlKOCTeA, HMeIOl.llHX Many10, HO KOHetI-
HYIO BSl3KOCTb, OT H.neaJibHblX :IKH.!1KOCTeH, HMeIOl.llHX HyJ1esy10 BSl3·
KOCTb 2). Ha 3Toro, no cymecTBy, cne.nyeT, '!TO yTsep>K.neHae Jla-
rpaH:ma (CM. npHM. 2 Ha CTp. 16) MO>KHO no.11npaBHTb, OOCTaBHB
«Hasbe - CTOKC> BMecTo «3iinep».
Eho yTBep>K.neuHe 6y.neT KpHTHtleCKH pacCMOTpeHo s rn. II;
OHO, no>KaJiyH, B npHHUHne sepHO .llJISI HeC31CUMaeMoeo BSl3KOro Te·
qeHHSI. 0.nHaKO, Mbl OOJiaraeM, qTo eCJIH OOHHMaTb ero 6yKBaJibHO,
TO OHO MO:>KeT BBecTH s aa6Jiy>K.neuee, OOCKOJlbKY SIBHO He Bbl.lle-
JieHbl nepet1HcJ1eHHble Bblllle npas.nono .no6Hble rnnoteabl H He y11TeH
TOT ymep6 s crporocrH, KOTOpblH o6ycJIOBJieH HX npHMeHeHHeM.
TeM ue MeHee HaM ue H3BeCTHO HH o.nuoro cnyt1asi, Kor.na
.ne-
.llYKUHSI, CTporasr KaK tpH3H'lecKH, TaK H MaTeMaTH'leCKH, npHBeJia
6bl K HenpaBHJlbHOMy aaKJIIO'leHHIO, HO JIHlllb O'leHb HeMHOrHe Bbl·
BOJlbl TeO{!eTH'leCKOH rH.llp0.llHH8MHKH MoryT 6b1Tb CTpOro ycTaHOB·
JieHbl. .llJIH caMblX HHTepeCHblX H3 HHX lllHPOKO HCDOJlb30BaJIHCb
OJJiHa HJIH HeCKOJlbKO H3 ynOMHHYTblX rnnore3 (A)-(F).
~ho MO>KHO noKa3aTb Ha npHMepe ypasHeHHH Hasbe - CToKca.
0HH SIBHO HenpHrO.llHbl .llJISI ylleTa peJISITHBHCTCKHX 3tptpeKTOB, MO·
JieKynsipuoA crpyKTyphl, KB8HTOBblX 34>tpeKTOB, pasuo KaK T8KHX
cneuHtpH'!eCKHX SIBJleHHH, KaK HOHH38UHSI, 3JleKrpocraTHtleCKHe CH·
Jlbl, 3arpS13HeHHSI BO B3BeCHX, KOH.lleHC8UHH H T. n •• K8:IK.llOe H3 KO.
TOpblx MO>KeT Bhl3B8Tb cepbe3Hble OCJIOlKHeHHSI, KaK 6y.neT
1) HinsheIwood C.; uerepye!ll no JlaArxenny [LIghthi11 M. Y.,
Nature, t78 ( 1956), 3431; CM. raK:lKe 111], r. 1, Bae.tteaae.
1
) CM. [3], § 1, 14: [Ill. r 1, Bee.tteuae; r. 2 . Bae.neaHe; Hunter Rouse,
Fluid Mechanics for Hydraulic Engineers, McGraw-l{ill, 1938, crp. 10,

18
r A. I. llapadOICCbl HeBJl3/COi!Q Te'teHUJI
~~- ~ ··------------------------
noKa3aHO HH>Ke. CTaJIO 6hITb, y>Ke cpaay WHpOKO HCilOJlb3yeTCH
rnnoTeaa (A). B cny11ae c:HCuMaeMocO re11eHHH ocTaercH OTKpbITbIM
.lI.a>Ke aonpoc O TOM, KaKOH CMbICJI HMeeT IlOHHTHe «BTOpOH> 8513·
KOCTH (§ 22, 33).
MbI He HacransaeM Ha TOM, l!T06bl snpe.n.b He HCilOJib30BaTb B
reopeTH11ecKoA ru.n.po.D.HHaMHKe runore3bI (A)-(F) -
.n.a>Ke a •rnc-
TOH MaTeMaTHKe npas.n.ono.n .o6Hbie coo6pa>KeHHH urpaior 011eHh
sa>KHYIO poJib 1). B ru.n .po.n .uHaMHKe npo.n.sn>KeHue e.n.sa JIH 6bIJio 6b1
B03MO>KHO 6e3 WHpOKOro HCilOJlb30B8HHH raKHX npas.n.ono .n.o6HblX
rnnore3, a fiOJIHaH CTporOCTb pe.n .Ko 6bIBaeT .ll.OCTH)KHMOH. Mbl TOJlb-
KO HacrauaaeM Ha TOM, 11ro, npe>K.n .e 11eM c11urarb Hay11Ho ycraHo-
BJieHHbIMH 3aKJIIOtieHHH, OCHOBaHHbie Ha npaa.n.ono.n.o6Hb!X coo6pa-
}KeHHHX, HX HaJI.o npOKOHTpOJIHpOB3Tb AU6o c noMOIUbIO crporux
.D.OKa3aTeJibCTB (KaK B 4HCTOH MaTeMaTHKe), AU6o c noMOIUbIO 3KC-
nepHMeHTa.
HanpOTHB, Mb! C4HraeM, 4TO HY)KHO TOJlbKO npHBeTCTBOB3Tb OT·
KpbITHe rn.n.po.n .HHaMHtieCKHX napa.noKCOB, HCKpeHHe npH3HaB He-
cnoco6HOCTb cymeCTBYIOIUeH MareMaTHKH ( H JIOrHKH) a.n.eKB3THO
OTo6pa>KaTb CJIO>KHbie H y.LI.HBHTeJibHble HBJieHHH npHpO.D .bl . OnbIT
noKa3bIBaer, '!TO 11eJioseqecKoe aoo6pa>KeHHe ropa3.n .o 6oJiee orpa-
HHtieHo, 11eM pecypcb1 npupo.nhl; KaK nucan nacKaJih, «aoo6pa>Ke-
HHe cKopee ycTatter nocrurarb, 11eM npnpo.n.a nocraBJIHTb>.
B CBH3H c 3THM, OCT3JlbH851 11acrb nepBOH rJiaBbI 6y.n.eT nOCBH·
w.etta attaJIH3Y HeK0Topb1x napa.n .oKcoa KJiaccu11ecKoif rn.n.po.n .ntta-
MHKH. B rJI. II Mbl y.neJIHM BHHMaHHe aHaJIOrHllHbIM (HO He CTOJlb
WHpOKO H3BeCTHblM) napa.n .oK'CaM «coapeMeHHOH»
. D.HH3MHKH
>KH,[I.KOCTeH.
§ 3. YpaeueHHSI 3iinepa
Mb! Ha4HeM C paccMOTpeHHH OCHOBHblX ypaBHeHHH .LI.JIH HeBff3·
KHX )Kff.ll .KOCTeH, BbIBe.n .eHHblX 3rtnepOM u JlarpaH>KeM. nycTb u =
= u(x, t) 03HattaeT BeKTOp CKopocnr >KH.D .KOCTH B TOl.fKe X B MOMeHT
speMeHH t. nycTb p(x, t) 03HatJaeT nJIOTHOCTb >KH.D.KOCTH, g(x, t) -
BHeurnee rpaBHT8U.HOHHOe 2 ) none H p (x, t) - .n.aBJieHHe B }KH,!J.KO·
CTH.
EcnH npHHHTb npaa.n .ono .n .06tty10 rnnore3y (E) H3 § l (urnopn-
PYH MOJieKyJIHPHYIO crpyKTYPY sew.eCTsa!), To nerKo noKa3aTb, 11ro
3aKOH coxpaHeHHff MaCCbl 3KBHBaJieHTeH cne.n .yiomeMy ypaBHeHHIO
1) CM. no Aa .[{ .. MateMataKa a npaB.llOnO.llOCiHble paccy>KJJ.eHHR, HJI, M. ,
1957. Dapa.n .oKcbl so:iHHKa!Ot JJ.a>Ke s 111tctoA MateMatHKe; CM. Northrop E. P .,
Riddles in Mathematics, Van Nostrand, 1944 .
2) 3rnM tepMHHOM aerop noJJb3yeTCR B 6011ee WHPOKOM CMblCJJe, qeM 9TO
06bl1!HO npHHRTO, npHMeHHII ero .llJJll nom1 MaCCOBblX (HJIII o6beMHb1X) CHJI. -
llpuM. nepee.

§ 3. YpasHeHu11 !Ja..epa
19
B '18CTHblX npOH3BO.llHblX:
div (pu) + !; = O (ypaeueuue uepa3pb1BHOCTH).
(1)
EcnH o6o3HaqHTb «cy6cTaHUHonan&uy10> npoH3BO.llHYIO no epe-
MeHH .llJISI ua6JIIO.ll8TMff, JlBH}l{ylUefOCff BMeCTe c >KHJlKOCTbIO, qepe3
D/Dt = o/ot + "f.u,.o/axk, TO MO>KHO nepenHC3Tb ( 1) B BH.lle
g; +pdivu=.0 .
(1 ')
Cnyqa10 nec~u.Mae.Mocru cooTseTCTBYeT Dp/Dt = 0, H, cne.lloea-
reJI&Ho, div u = 0.
npu U = 0, K0f.ll3 >KH.llKOCTb H3XO.llHTCff B COCTOHHHH nOKOR,
uanpH>KeHHe B )KffJlKOCTH Ha JII060H 3JieMeHT nosepxHOCTH )leH. ·
CTByeT no HOpMaAU K HeMy. 3TO- <j>H3HlfeCKOe onpe.aeneHHe
>KH.!lKOCTH; 3KcnepHMeHT8JlbHO nposepeHO, qTo eMy Y.llOBJleTBOpSl!OT
MHOrHe peaJI&Hble eew.ecrea .
3iinep npe,llDOJIO>KHJI, qro 3TOT 38KOH rH.llpOCT8THKH npuMeHHM
T8K>Ke K .llBH>KYUJ.HMCSI )KH. , llKOCTffM, T . e . B rH.lJ.PO.llHHaMHKe. 3ror
S8KOH npH6JIH>KeHHO Y.llOBJleTBOpHeTCff 80 MHOrHX cJiylfaffx .llBH>Ke-
HHff >KH.llKOCTeH (HCKJIIOqaSf o6JiaCTH e6JIH3H rpaHHUbl). HanpHMep,
H3MeHeHue CKOpOCTH Ha 50 M/ce,c B CJIOe 803.U.yxa TOJIW.HHOH B qer-
eepTb MHJIJIHMerpa Bb13bleaer ycunue CJlBHra, cocraensi10w.ee npu-
Mepuo 1/2000 arMoccpepuoro .l{aeneunH ([3], crp. 2).
HenpepblBHble JKH.ll .KOCTH, yJ1.oeneTBopsi10mue runore3e 3iinepa,
uasbl"BaIOTCH He8Jt3KUMU 1). KaK noKa3aJI KowH, Hanpst>KeHue e ue-
BJl3KoA }KHJlKOCTH .ll .OJIJKHO 6b1Tb OJJ.HH8KOBblM BO scex uanpasne-
HHffX (H30TPODHhlM); nonyqa10w.~S1Cff CK8JISIPH8JI <l>YHKUHff p (x, t)
MOJKeT 6b1Tb Ha3BaHa aaeAeHUeM. Jl.anee, 38KOH coxpaHeHHff KOJIH·
' lecTea .llBH}l{eHHff 3KBHB8JI0HTeH CJieJJ.yIOW.eMy BeKTOpHOMy ypaB·
HeHHIO B '18CTHblX npOH3BOJJ.HblX:
Du1
Dt + P grad p = g (ypaeueHHe JJ.BH>KeHHH).
(2)
.ll.nff roro qro6bl DOJIY'IHTb 38MKHYTYIO CHCTeMy ypasueHHH B
qacTHblX npOH380.llHhlX, B KOTopoii Bee npOH3BOJJ.Hble no BpeMeHU
11ory1 6b1Tb B blpaJKeHbl qepe3 npoH3BO.ll.Hble no npocrpaHcTBeHHblM
,coopouHaTaM 2)' K ypaeueHHSIM ( 1)' (2) HYlKHO .u .o6aBHTb ew.e OJlHO
COOTHOWenue. B reopeTH'leCKOH MeXaHHKe O.lJ.HOPO.llHhlX HeBff3KHX
1) 8 COOTBeTCTBHH C rHROTe3oA (8) H3 § I 803.llYX H BOJIY MOJKHO paCCM8·
TpH88TI> K8K HeE113KHe lKIIJlKOCTH.
1) TaK, '!TO Ha'laJU,Hble ycnoe1111 6yJlyT onpeJ1en11Tb aua11y KowH e 061,111uou
M8TCM8Tll'lecKOM CMblCJJe,

20
f A. /. llapaOOKCbl He8Jl3K020 tel/eHUfl
iKHJlKOCTeH o6bllJHO BBO.ll .HTCH COOTHOWeHHe, CBH3hlB810W.ee IlJIOT·
HOCTh H )laBJieHHe:
p = h (p) (ypaBHeHHe COCTOHHHH).
(3)
Ba por pon Hhie TetJ e HHH. HeeH3KHe mHJI.KOCTH, y.n.oene-
reopn10mHe ycnoa·H10 (3), Moryr 6bITh tta3BaHbl 6aporponHhIMH,
a .ll.BH)KeHHH )l{H.ll.KOCTH , Y .ll .OBJieTBOpHIOI.I . .lHe ypaBHeHHHM ( I )-(3) ,-
«6aporporrHhlMH retJeHHHMH». 3rH retJeHHH ecrpetJaIOTCH a (rrpH·
6JIH)KeHHO) O)lHOPO.LI .HhIX )l{H)J.KOCT .HX
npH YCJIOBHHX, KOTOpbie
S!BJIHIOTCH repMO)lHHaMHtJeCKH o6parHMhIMH. (flo)l «O)lHOpOJlHOH
)KH.ll.KOCThlO» Mbl ITOHHMaeM )l{H)lKOCTh, HMe10my10 O.ll .HOpOJlHOe crpoe·
HHe, HarrpHMep tJHCTYIO BO.ll.Y HJIH B03JJ .yX.)
l1MeHHO raKHe )KH.ll .KOCTH o6hilJHO paccMarpueaIOTCH e aKyCTHKe
II B a3p0.1lHHaMHKe 6oJihI.IJHX CKopocreii. Bhlcrpoe cmarHe H pacI.IJH·
peHHe - THITHlJHhle a)I.Ha6aTHlJeCKHe rrpouecCbl 1) a TOM CMhlCJie, lJTO
Momtto rrpeHe6pe% rerrnonpoeo)lHOCThIO. KpoMe roro, rrpette6pe)Ke-
HHe rerrnorrpoao.11.HocTblO norwiecKH He rrporttsopeqHT npeHe6pe}Ke-
HHIO BH3KOCTblO B ypaBHeHHH (2). ITOCKOJlbKY K8K rerrnorrpoBO)l·
HOCTb, raK 11 BH3KOCTh rrpeJI.craBJISIIOT co6oii MOJieKyJIHpHble HB·
JieHHSI.
B cnyqae H)leaJihHOro ra3a c repMO.ll .HHaMHqecKHM ypaatteHHeM
cocTOHHHfl p = pRT H rrocroHHHblM orHoI.IJeHHeM rerrnoeMKocreii
Cp/Cv = i 3JieMeHrapHhie paccym.n .eH1rn )laIOr JlJIH a;u1a6arHqecKoro
ret1emrn roornoI.IJem1e
p= kpT
(3a)
-
TaK H83blBaeMoe nO.llUTpomwe ypaaHeHHe COCTOHHHSI .11.JIH H)le-
aJihHOro repMoJI.HHaMHqecKH coaepI.IJeHHoro ra3a. Tipe.n.enhHhIH cny-
qajj 1 = I cooreercrsyer H3orepMHtJecKoMy retJeHHIO (6ecKoHetJHaH
TerIJIOeMKOCTb HJIH B 6eCKOHeqHoM H3orepMHtJeCKOM pe3epeyape 6ec-
KOHe4H8H rrpoBO)lHMOCTb).
YpaBHeHHe (3a) .ll.OCT8T04HO T04HO )J.JIS! MHOrHX 38)laq ra30BOH
)lHH8MHKH; )lJISI B03)lyxa i = 1,4. 0)lH8KO )lJIH }l(H)lKOCTeH ypaBHe-
HHe (3) He06XO)J.HMO 6parh (rrpH6JIH.>KeHHO) e BHJJ.e (p - p, ,) = kp 1 ,
r)le Pv eCTb )l3BJieHHe napos rrpH KaBHT8UHH (CM. § 42).
CooTHOWeHHe BH)J.a (3) HBJIHeTCH T8K)l{e rrpHeMJieMblM )lJIH )l{H)l·
KOCTeH, KOTOpble TOJibKO He3Ha4HTeJibHO C}l(HMaeMbl (r. e. rrpH CKO·
1) HanoMHHM,
' !To HhlOTOH («Principia Mathematica», KHHra II, OTJJ.eJJ 8,
npeJJ.JJO)KeHHe 48; pyccKHH nepeBOA- B «Co6paHHH TPYJlOB A. H . KpblJJoBa>,
T. VII, M-JI., 1936; CM. TBM )Ke, C 'lp. 480) npHHHMa,, )lJJH H30TepMH'leCKoro
Te'leHIIH 33KOH BoiiJJH,
' !TO npHBeJio K 11enpaBHJlbHOMY BblBOJlY CKOpocrn 3BYK8.
Ourn6Ka Hbt0Totta 61,1na ttcnpaa.,eHa JlanJJacoM ([7], crp. 477; s pyccKoM H3Jl8·
HIIH crp. 596; CM. TBK>Ke yKa3aHttb1ii TOM «Co6patt11H rpyJlOB A. H . Kp1,1noaa»,
crp. 485, np11M. 175).

§ 4. nore/lljUQA CKOpOCTU
21
pocrnx ropa3JlO MeHbllIHX cKopocrn 3ey1<a, oco6eHHo B o6b!lJHbJX
.>KH)lKOCTHX). B 3TOM CJiy'lae MO.>KHO npocTO IlHCaTb
P=Po
(36)
H fOBOpHTb 06 oa1wpoa1-tou HeC:HCUMaeMOU He8fl3KOIJ. :HCUaKOCTtl. 0 .Ll·
HaKo B 3TOM CJiylJae y>Ke HeJib3.H Bb!pa3HTb Bee npOH3BO)lHble no
BpeMeHH 4epe3 npoH3BO.l].Hbie no npocrpaHCTBel!HbIM KOOp..QHHaTaM.
§ 4. floTeHUHaJI CKOpOCTH
OcHOBHbie ypaaHeHHH 3ib1epa ( l )- (3) n03BOJHIIOT noJiylJHTb
pa3JIHlJHbie cpyHJJ.aMeHTaJibHb!e CJieJJ.CTBHH , HMelOW.He 11rnoro Ba}K-
Hb!X npHJio>KeHHH.
CaMbIM cym.ecrneHHbIM CJieJJ.CTBHeM HBJIHeTC.H TeopeMa f eJibM-
rOJibUa, cnpaae,a .JIHBa.H AJI.H 6aporponHoro Te'leHHH B KOttcepBaTHB-
HhlX rpaBHTaUHOHHbIX nOJIHX (T. e. npH g = -VG). 3ra reopeMa
([7), CTp.54;[1*)1)),T. l,cTp.149) yrnep)K,a.aer1rnaapHaHTHOCTbU.Hp-
KYJIHU.HH r =~~Uk dxk no JII060MY 3aMKHYTOMY KOHTYPY, JJ.BH -
>KyUJ.eMycsi BMecre C )KHJlKOCTblO, T. e. BO BCHKHH MOMeHT apeMem1
COCTOHW.eMy H3 OAHHX H Tex }Ke lJacrnu )KHJJ.KOCTH . CJie,a.oBaTeJibHO,
eCJIH B HalJaJibHbIH MOMeHT )KH.IJ.KOCTb HaXOJUHCH B noKoe (Hanp!i-
Mep, Bb!TeKaeT H3 Heno,ll.BH)l{HOro pe3epayapa) H. eCJIH KOHtyp
ocraetCH ace spel\rn 3aMKHYThIM, To UHPKYJIHUIIH acer.na JJ.OJI)Ktta
paBHHThCH HYJIIO. 3TO 3HalJHT, lJTO .11.0JDKeH cym.eCTBOBaTb JIOKaJihHO
0!11H03HalJHblH CKaJIHPHhlH noTeH4UaA CKOpOCTU U (x, t), T. e. TaKaH
CKaJisipHaH cpyHKUHH TOlJKH, lJTO
u (x, t) =VU= grad U.
(4)
Te4eHHH, 06J1a.n.a10m.He taKHM CBOHCTBOM, Ha3bIBalOTCH (JIOKaJib·
HO) 6e3euxpe8b(.MU 2). CJie.D,OBaTeJihHO, B oaHOC8fl3HOU o6JiaCTH, Ta·
KOH, KaK o6JiaCTb BHe HeKOToporo TBepJJ.OfO TeJia a npocrpaHCTBe
liJIH fiOJIOBHHa CHMMeTpH4HOH o6JiaCTH BHe Kpyroaoro UHJIHHApa Ha
nJIOCKOCTH, CKOpOCTh u .11 .0JDKHa 6b!Tb 0.ll.H03HalJHOH cpyHKUHeH BO
sceu o6JiaCtH.
B cJiy1Jae 6aporponHbIX re1JeHHH nptt orcyrctBHH BHel.lIHHX rpa-
BHTallHOHHbIX CHJI )l.JIH 6e3BHXpeeoro )lBH}KeHHH [T. e. eCJIH BhlnOJI-
HSleTCH ypaBHeHHe (4)] MO)l{HO noJiylJHTh HHterpaJI ypaBHeHHH .11 .BH -
>KeHHH, TaK Ha3b!BaeMoe ypaBHeHHe BepttyJIJIH
Idp
1
au
-P =P(t)- 2 VUVU - at' p=h(p).
(4')
1) 3Be3JlQqKa o603Haqaer CCblJIKY Ha JlOnOJIHHTeJibHYIO JIHreparypy. -
flpuJt.
pea.
2
) B pyccKoii nttreparype qaw.e Hcnonb3yeTrn TepMHH «norettu.11anbH1,1e T('
' !eHHR>. -
llpUJI. nepeB.

22
I'A. I. llapaaotcCbl HeBJl3K020 retteHUR
neikTBHTeJibHO, ypaBHeHHH .llBH}KeHirn ( 6e3 rpaBHT3l.U!OHHOf0
CJiaraeMoro) npe.n:cTaBJHIIOT co6oii B TOtJHOCTH rpa.n:HeHT COOTHOIIIe-
HHH (4').
Hec}KHMaeMble Teq eHHH. B cJiyttae o.n:Hopo.n:Hhrx HeC:>K:II·
MaeMbIX :>K:ll.llKOCTeii MO:>K:HO o6o6mHTb ypaBHeHHe BepHyJIJIH ( 4')
T8K, tJTOObI YllHTb!BaJICH 3<p<peKT rpamnau.i,m . .UeHCTBHTeJibHO, )J,JIH
6e3BHxpeBbIX HeC}KHMaeMb!X Teqerrnii rpaP,HeHT COOTHOIIIeHHH
{
1
au
}
P=P(t)- Po 2 \7U\7U+at+0
(5)
3KBHBaJieHTeH ypaBHeH!rnM .llBHIB:eHHSI C rp 8BHT8UHOHHbIM tJJieHOM.
BoJiee Toro, B aroM cJiytJae ypaBHeHHe ( 1} cBOJJ:HTCH I< BH.llY
div u = 0, OTKyAa noJiyllaeM ypaBHeHHe
'v
2U=O.
(6)
HaKoHeu, otteBHAHO, tJTO Ha JII06oii HenpoenuaeMofi TBepaofi
rpaHHUe npOH3BO,lJ,HaH
au
an =F(x, t)
(7)
onpe.n:eJISieTCH HOpMaJlbHOH CKOpOCTblO .QBH}KeHHH ~TOH rpaHHllbl •
.UmI O)J.H03H8 1!HbIX BO Bceii o6JiaCTH <pyHKUHH u (x) ypaBHeHHH
(6) H (7} npe,llCT8BJIHIOT KJI8CCHtJeCKYIO 38,llaqy TeopHH IlOTeH-
UHaJia, TaK tta3bIBaeMyro 3aiJatty HeuMana. KaK MbI ysn.n;HM a § 5
H rn. VI~ 3ra 3a.n:a4a HMeer 60JibllIOe 3HatJeane JlJIH reopeni:ttecKott
rH,llpOJ]JIH3MHKH. Ho rrpe:>K:P,e OTMeTHM, tJTO 3,lleCb rro.n:pa3yMeaaeTCH
BbmOJIHeHHOH nmore3a (F) H3 § l: npe.n:noJiaraeTcH, l!TO 3a.n;a11a
HeiiMaHa JJ.0JI}KH8 HMeTb O,llHO II TOJibKO O)J.HO O)J.H03H8tJHOe perne-
Irne u (x, t) )].JUI pa3yMHbIM o6pa30M onpe.n:eJieHHbIX rpaHHII.
CTpttMe'laTeJibHO, t{TQ JJ.JIH CTpororo .llOKa3aTeJibCTB8 3TOro Mare-
MaTll'IeCKOfO npe.n:noJIO}KeHHH, B03HI1Killero ll3 fHJJ.POJJ .HH3MHtJeCKHX
paccMOTpeHHH, rrorpe6osaJiocb 6oJiee tJeM 50 Jier. B HacTOmn.ee
BpeMH 3TO OCHOBH8H TeopeMa o6meii: TeopHH IIOTeHUHaJia ([4],
cTp. 310 -311; [2*]).
9Ta TeopeMa IIOK83bIBaer, tJTO ecJIH HeC}KHMaeMaH HeBH3K3H
:>K:HJJ.KOCTb B H8tJ3JlbHhiii MOMeHT HaXO.llllTCH B COCTOHHHH IIOKOH, TO
IIOJie CKopocTeii B JII06oii: MOMeHT BpeMeHH 38BHCHT TOJlbKO OT
M.2HOBeHIWU CKOpOCTH rpaHIHJ;bl II He 33BHCHT OT npe)J.IIIecTByIOlllHX
COCTOHHHH. CTpHBe)J.eHHbie TeopeMbl IlOK33blB3IOT T8K}Ke, tJTO JIBH·
}KeHHe JIIOOOH tJ3CTH rpaHHl.IbI MfHOBeHHO OK83bIB3eT B03.QeHCTBHe
Ha Beeb 061,eM )KHJJ.KOCTH: CKOpOCTb CHrHaJia paBHa 6eCKOHetJHOCHf
(3TO cornacyeTC.H H C cpH3H4eCKOH HHTYHIIHeii).

§ 6. CTa1,1uoHapHble 6eaBuxpe8bte re~eHu11
23
§ 5. CTRU.HOHapHble 6e3BHXpeBble Te11eHHfl
CJiyqafi CTaU.HOHapHblX (mIH ycraHOBHBlliHXCB) re11eHHH, KOf.lla
u = u (x), O'leBHJI.Ho, HMeer oco6oe atta'leHHe. Hcxo.zu1 H3 pesyJib-
raros § 4, rHJI.PO.llHHaMHKH XIX BeKa c 1mTaJIH npaB.llono,JJ .06HI>1M,
'!TO
JJ.JI}I TBep]J.oro TeJia, npoXOJJ..HllI.ero c IlOCTOHHHOH CKOPOCTbIO
.llOCTarotJHO 60JibIIIOe no cpasHeHHIO c ero pa3MepaMH paCCTO.HHHe
B HeorpaHHtJeHHOM OO'beMe iKHJJ.KOCTH , BH3KOCTb KOTOpofi ,JJ .OCTa-
TOtJHO MaJia H KOTOpaH nepBOHatJaJibHO HaXOJJ.HJiaCb B IlOKOe, MOiKHO
HanHCaTb paBeHCTBa:
U=U(x), p=p(x), g=g(x),
H T. JI..
-
JJ.JIH oceii KOOPJJ.HHaT, :iKeCTKO CBH3aHHbIX C reJIOM, OTHO-
CHTeJibHO KOTOpb!X iKHJlKOCTb JI.BIDKeTCH c IlOCTOHHHOH CKOpOCTblO a.
5lcHo, 'ITO TaKHe npaBJJ.OilOJI.OOHble BblBOJl.bI OCHOBaHbI Ha rnno-
reae (B) § l.
Ecmi: HCXO.llHTb H3 3THX npaB,ll0IlO)l06HbIX :'laKJIIOtJemi:fi, TO ii:aJiee
MOiKHO .neikTBOBaTb CJieJJ.yIOlll.HM o6pa30M. )lJIH CTaUHOHapHbIX Te-
tJeHHH npH U = U (x) ypasHeHHe .llBHiKeHHH (2) IlOCJie O.llHOKpar-
HOro HHTerpHpOBaHHH no npocrpaHCTBeHHblM KOOpJJ.HHaTaM CTaHO·
BHTCH 3KBHBaJieHTHbIM ypaBHeHHIO 1)
2
1
vuvu+ f d: +a=::, HJIH l)uidu1+ h~;) +dO=O. (8)
9TO ypasHem1e Ha3hIBaerrn ypasttemieM BepHyJIJIH JJ.JIH crau.tto·
HapHoro TetJeHHSI; B CJiyiiae HeCiKHMaeMOH iKHJI.KOCTH OHO npIIHH-
Maer H3BecTHbIH npocrott BII.n::
P=Po-Po ({ VUVU + a).
(8*)
no.11 .06HhIM o6pa30M ycJIOBHe roro, 'ITO CKOpOCTb TeJia OTHOCII·
TeJibHO iKH.llKOCTH Ha 6eCKOHe1rnOCTH paBHa -a, MO}KeT 6hJTh 3ailII-
caHo B BH.ne
Jim grad U =a
x~oo
(9)
H .llJIH HeC}KHMaeMoro, H .llJIH CiKIIMaeMoro retJeHmI.
HaKoHen:, nocKOJihKY retJeHIIe CTan:uoHaptto, TO AOJiiKHhI OhITh
cran:iroHapHbI tt rpaHHU:bI TetJeHHH. 0TCIO,[J.a ycJIOBHe HenpoHHU.aeMo-
CTH (7) CBOJI.HTCH K YCJIOBJUO
au_
0
an-
Ha rpamHJ,e.
(7*)
1
) EcJJH cnpaBe,D:JJHBO COOTHOllleHHe (3a), TO f dp/p == lP/(1- 1) ~ =
== c2/(1- l).

24
FA. I . flapaiJ01wbt neB!l3Koeo rettenu1t
B cJiyqae 6e3BnxpeBoro C}KHMaeMoro TetJemrn ypaBHeHne aepa3-
pbIBHocrn ( l) Bee eru.e MO}KHO 3arrncaTb np:n IlOMOW.H e,QHHCTBeHHOH
HeH3BeCTHOH <PYHKUHH U(x), ecAU TOJ!bKO npeHe6pet!b 9Q)<peKTOM
rpaBHTar.urn , tJTO OObitJHO ,QOIIYCTHMO npH AOCT3TOtJHO OOJibllIHX CKO-
pocrnx, Kor,n;a CTaHOBHTCH 3aMeTHOH C}KHMaeMOCTb 1). (ECJin rpaBn:.
TaUHeii HeJib35I npette6petJb, KaK, Hanp:nMep, B cJiyqae aTMocqiep-
HbIX ,ll .BH}KeHHH OOJibllIHX MaCIIITaOOB, TO ycJIOBHe (9) He MO}KeT
Oh!Tb BbI!IOJIHeHO, ,113}Ke HeCMOTPH Ha TO, tJTO 6e3BHXpeBoe TetJeH:ne
HB.' rn eTCH AOilYCTHMbIM.)
K:nHeMaT:nKa 6apoTponHoro TeqeHnH. noJiaraH
G = 0 B ypaBHeHHH (8), np:n OIIHCaHHhIX BbIIIIe ycJIOBllHX MbI MO-
}KeM IIOJIYtJHTb paBeHCTBO
dp
2
1
dp=C = h'(p)
1
(9*)
r JI.e J - ¢YHKUHH , o6parnaH qiy1mu1m 2pof Po - 2 f dp / k'(p) p* 2).
c JJ.pyroii CTOpOHbl, H3 ypaBHeHHH (I) np:n {)p/ot = 0 CJie)J.yeT
p-1 div (pu} = 0, HJI:n
(10)
L(pyraH ¢opMa ypaBHeHHH ( IO) HMeeT BHJJ.
(lO*J
rJI.e JioKaJibHoe «tJHCJio Maxa» M
q/c ecTb OTHOIIIemre JIOKaJibHOH
CKOpOCTH Te'- !eHmI q K JIOKaJibHOH CKOpOCTH 3ByKa c, a Bee K03ql<pH·
UHeHThI UjUk/q2 MeHbIIIe HJIH paBHbI l.
no11.cTaBHB B ypaBH€HHe ( 10) Bbipa}KeHHe JI.JIH 1/c 2 , B35ITOe H3
qiopMyJibl (9*), Mhl IIOJIY'- IH M ([IO], crp. 240) ypaBHeHHe
(11)
011.HO-e,QHHCTBeHHOe ypaBHeH:ne B '- !8CTHhIX rrpOH3BO,l(HbIX ( 11) BMe-
CTe c KpaeBblMH ycJIOBHHMH (9) H (7*) CBO,l(HT 3a,naqy .!l.JlH CJiy4aH
CTaUHOHapHoro CJICU.!ltaeM.020 Tet.JeHHl{ 6aporporrHOH }KHD.KOCTH HY·
JieBOH (MaJioH?) BH3KOCTH K ~pyroii npaB)J.OflO,l(OOHOH KpaeBOM 3a·
1) B c.11y•rne HeClKlU,taeMOCTH rpaBHT3UHOHHhlH 9<p<peKT CBOnHTCS! K Ot5b!KHO·
BetlHOH rnnpocraTH'leCKOii non'heMHOH cirne, KaK TIOK333HO B § 21.
2) DpH 9TOM TIOCJle,llH!OlO <PYHKUH!O ttano paccMaTpHB3Tb KaK cpyHKUl!lO apry·
MPHTa h'(p). -
llpu.11 . nepee.

JI.aqe. EcJIH TOJihKO nocJieJI.H5151 3aJl.atJa pernena, TO H3 ypasHe1m,1
( 8) MO:lKHO JierKo Ha iiTH TIOJie .[l3BJiemrn.
TaKHM o6pa30M, Mhl CBeJIH 38.[l3lJY CT3UHOHapHoro TelJeHHH K
lJHCTO KHHeMaTHlJeCKOH 38.[lalJe. EcJIH .uaHo mo6oe MaTel\laTHlleCKOe
perneHHe ypaBHeHHH ( 11), (9) H (7*) H eCJIH nocpe,!lCTBOM ypaBHe-
HHH (8) orrpe.ueJieHO TIOJie .[l3BJieHH51 npH G = 0, TO ypaBHeHHe Jl.BM -
)KeHIHI (2) y.uoBJieTBop5IeTCH aBTOMaTHlJeCKH. OtJeBH.!lHO, LJTO 3a-
JJ.Ha Heii.MaHa H3 § 4 noJiytJaeTCH KaK npe,UeJibHbIH cJiy'latt rrp H
c--+ - oo . Llorryw.eette (F), TaKHM o6pa30M, rr03BOJI5leT rroJiy'IHTb ro-
paa)J.o 6oJibllle, a HMeHHO, LJTO perneHHe MO:lKHO pa3JIO:lKl1Tb ITO CTe-
rreH5lM M2 (MeTOJJ. P3JieH - 5:Ieueea, [15], cTp. 275).
§ 6. Uapa)J.OKC o6paTUMOCTH
OJ.I.HOH H3 cpyHJJ.aMeHT8J1hHb!X 33.[laLJ rH)J.pOMexaHHKH 51BJI5leTC51
onpeJJ.eJieeee CHJibI,
.u ettCTBy10w.ei1 ua TBep)J.oe TeJio, Haxo.uHw.eeCB
B CTaUI-IOHapHOM rrocrynareJibHOM )J.BH:lKeHHH c TIOCTOHHHOii CKO-
pOCTblO a B 0)1.HOpOJI.HOH IIOKOHW ,eii.CH )KH)J.KOCTH . EcJIH TBep.uoe
TeJIO Jl.BH)KeTCH rrapaJIJieJlbHO HeKOTOpoii IIJIOCKOCTH Cl!MMeTpHH, TO
3TY CHJiy MO:lKHO pa3JIO:lKHTb Ha JI060BOe corrpOT1IBJle,HHe D, TIO.[lb-
e~IHYIO CHJIY L H MoMeHT M, .1 .Leiicrny10lll;Hii B 3TOH rrJiocKocrH.
JlarpaH)K MOr 6b! 33MeTHTb, ITOJ!b3Y51Cb BeCbMa rrpOCTb!MH coo6-
pa:iKeHH51MH o6parUMOCTU, lJTO HJl.eaJIH3HpoBaHHa51 KpaeB351 33Jl.348
§ 5 MO)KeT He rrp1rnecn1 K rrpaBHJibHOMY pe3yJibTaTy rrpH orrpeJl.eJie-
HHH corrpoTHBJieHH5l, HCIIblTbIBaeMoro peaJibHblMH TBepJl.blMH TCJl3MH
nprr )J.BH)KeHHH B peaJihHb!X :lKHJl.KOCT51X . 0cHOBHa51 Mb!CJ!b 3aKJIIO-
qaeTC51 B CJie.[ly!Olll;eM ( CM. [l]).
Orr p e Jl. e JI e H He 1. 06paw,euue TeLJemrn u (x; t) orrpeJI.CJIBeTCH
KaK v(x; t) =
- u(x; -t), nptt4eM B o6oex reqemrnx !l:aBJieHee H
TIJIOTHOCTb B COOTBeTCTBYIOlll;HX TOLIKax O)J.HHaKOBbl.
CTpHMOH IIO)J.CTaHOBKOii: MO)Kl-10 TIOK333Tb, l!TO 06pame1-1rre JII0-
6oro TeLieHHH, YJJ.OBJieTBOp51!0W,ero ypaBHeHH51M ( 1)-(3), TaK>Ke
y,u;oBJieTBOpHCT ypaBHeHHHM (1)-(3), rrpaB.ua , rrpH o6paw.ermtt H
KpaeBblX ycJIOBHH. B l!3CTHOCTH, crrpaBe)J.JIHBa CJie)J.y!OW:351 JleMMa.
JI e MM a. EcAu u (x) ecrb cra4uouapuoe 6e3euxpeeoe TetJeuue
BOl~pyc T8epooc0 npenflTCT8Ufl U U ( oo) = a, TO TQ/W8blM RB./lfleTCfl
u v (x) = -u (x) npu v (oo) = -a. KpoMe TOW, nO.llfl oaB.!leHUfl,
TQK aice KaK u D, L u M, oauuaKOBbl O.llfl u (x) u v (x).
~ha JICMMa HaXO)J.HTC5l B KG1Jecree1-t1-tOM rrpOTHBopeLIHH c JJ.HHa-
MHKOii peaJibHbIX )KH)J.KOCTefi: B .n.eiicTBHTeJibHOCTH H3MeHeHHe Ha-
rrpaBJieHH51 ,llBH)KyW,erorn TeJJa Ha npOTHBOITOJIO)KHOe o6bJLJH0

26
I'A. J . llapaiJOKCbl He8113K020 Te'teHUfl
BJieqeT o6pame,HHe aeJIHtJHH D H L (xorn H He M) 1), a He ocTaB-
m1eT HX HeM3MeHHblMH.
DoytJHTeJihHO npoaHamrnHpoaaTb npeJI.b1,nymee nponmopetme
noJI.poottee. DoKa He ycTaHOBJietto, 4TO Kpaeaa51 saJ(aqa B § 5 KOp-
peKTHo nocTaBJieHa, HeJib35I JI.eJiaTb BblBO)l O TOM, lfTO ee ypaBHe-
HHH OlllHOOtJHbI. B03MO:lKHO, IITO noTpeoyeTCH BBeCTH KaKoe-HH6y,n:h
.n :o r rOJIHHTeJibHOe ycJIOBHe• .[(eikTBHTeJibHO, KaK Mbl YBHAMM a § l 0,
9TO MO:lKeT OKa381;'.bCH cnpaBeJ(JIMBbIM JI.JIH caepxsaytW80e0 TeIIeHHH
( T. e . ecJIH tIHCJIO Maxa M > 1). 4TOOhl noHCHHTb 9TO, npoae.rr .e ·M
cJie,n:y10mee paarpaHHtieHHe:
O II p e .n: e JI e H H e 2. By,rr,eM H83blB8Th rH,n:poJI.HHaMHlJeCKHe TeO-
pmI HenOAHblMU, ecJIH COOTBeTCTBYIOIII.He YCJIOBllH onpeJI.eJHIIOT 06-
reKaHHe J(aHHOfO npemITCTBHH He eJI.HHCTBeHHhIM o6pa3oM; nepeon-
peoeAeHHblMU, eCJIH 9TH ycJIOBHH MaTeMaTl!lfeCKH He COBMeCTHbI;
11,0;)ICHblMU, eCJill KOppeKTHO IlOCTaBJieHH85I 38.l(atJa J(aer rpyoo Olliff·
6otIHhie peayJILTaTbI.
Teo p e Ma 1. BcJUrn!I. o6paTUMaR 2uopoounaMuttecKaR reopuR
a o'fHomeHuu pacttera .1to6oaoe0 conporua.1teHuft u noo<oeMHou CU/l.bt
fl8A!l.eTC!I. HenoAHOU, nepeonpeoeAeHHOU UAU /l.00/CHOU.
J/.03BYKOBOU CAY,.WU . B
, !J:03ByKOBOM cJiyqae, 1\1 < l, no KpaiiHeH
Mepe ,n:JIH )lOCTaTOtJHO MaJioro lJHCJia Maxa HeJJ.aBHO 6h1JI0 noKa3a-
Ho 2), irTo KpaeBaH saJ(alfa, onpeJJ.emi:eMaH ypaBHeHH5IMH ( 11), (9)
H (7*) H3 § 5, HBJIHeTCH KoppeKrno nocraBJieHHOH. DocKOJibKY 9Ta
aa,n:aira 9JIJIHilTHlfecKoro nma, ee MareMarttirecKoe pemeHHe U (x)
)J.OJl:lKHO Ob[Tb aHaAUTUtteCICUM. Orcro,rr,a Mb[ 38KJIIOlfaeM, lfTO ypaB-
HeHHH 9i:iJiepa - JiarpaH:lKa J(810T /t0;)1CflY10 Teopmo JI.JIH CTaIJ.HO -
HapHoro ,ll03BYKOBOro IlOTOKa.
01CO.A038YICOBOU cAyttaii. no IIOBo,n:y 9Toro CJiyiraH, Kor,rr ,a B JJ.O ·
3ByKOBOM IlOTOKe HMeIOTC.H JIOKaJibHhle CBepxasyKOBble 30Hbl, Bbl·
cKa3aHo MHOro pa3JIHlfHbIX H IIJ)'OTHBOpeirHBhIX yTBepm:,n:eHlIH. BbIJIH
nocrpoeHbI MaTeMaTHlfeCKHe MO,LI.eJIH IlOJI.OOHbIX OKOJI03BYKOBbIX Te-
ireHHH 3 ), HO OHH, no-BHJJ .HMOMY , oireHh cJia6o orpaJKaIOT q>HaHirecKy10
1) l(JiaCCH'!eCK351 rH)lp0,ll,HH8MHK3 npaBHJlhHO npe.n:cKa3bIB3eT TeH,ll,em:i.mo OCC·
CHMMeTpHttHbll{ npenHTCTBHii TIO,ll,CT8BJlf!Tb TIOTOKY 6oJiee urnpoKylO CTOPOHy: cp. [7],
§ 71, 124. (B CJiy'lae Te.11 , 06Jia)lal01UHX npO,ll,OJ!bHOH CHMMeTpHeii, npH o6pamemrn
TIOTOKa L ocraercf! HeH3MeHHblM, a y M 113MeHHeTC51 3H3K.)
2) Graff i D., I. Rat. Mech. Analysis, 2 ( 1953), 99-106; Gilbar g D.,
raM .lKe, 233-251; Gilbarg D. and Serrin J., raM .lKe, 4(1955), 169-175;
Bers L., Comm. Pure Appl. Math., 7(1954), 441~504; Finn R. S. and
G i Ib a r g D., TaM .lKe, 10 (1957), 23-64 H Acta Math., 98 (1957), 265-276.
3rn o6o6maer peJyJihTar H3 § 4 Ha CJIYttaii M = 0 (3a)la'la HeiiMaHa). [CJiy'laii
MaJibIX M CM. B [3*] - llpuM. peiJ.J
3) M H 3 e c P., MareMaTH4eCKaH reopns Te'!eHHH Cllrn~rneMoH .lKH,J .KOCTll,
HJI, M., 1961, § 25, n. 3.

§ 7. llapaiJoKc l{aAaM6epa
27
KapTHHy. Eru.e 6oJiee ,llpaMaTH'leCKHM o6CTOHTeJibCTBOM HBJIHeTCSl
TO, l!TO JJ.Jlll HeI{'OTOpbIX npo¢mneii HHKaKoe OKOJI03BYKOBOe Tel!eHHe
6e3 y,llapHOH BOJIHhl HeB03MO>KHQ. 9TOT napa)lOKC OKOJI03BYKOBOro
TeIJeHHH He.n .aBHo ycTaHoBJieH K. Mopaaeu 1). no repMHHOJIOI'HH
TeopeMhl 1, 3TO 03Hal!aeT, 'lTO 3a,lJ.aIJa OKOJI03BYKOBOro TeIJemrn B
TeopemqecKoii (9iiJiepa - JlarpaH>Ka)
rn.n:pomrnaMHKe Mo>KeT
6bITh nepeonpeiJeAennoii.
B § IO Mhl yBHJJ,HM,
'I TO 3an;aqa cBepx3BYKOBoro Te'tleHHR-TH -
nHqHafI HeilOJIHaH 3a,n:aqa, H npHMellaTeJibHO, l!TO pa3JIH't!Hble pa3·
pemeHHfI napaJlOKca oopaTHMOCTH B Tpex npe,ZI .bIJlYIUHX cJiyIJaHX
HaXO)];.fITCfI B COOTBeTCTBHH c o6ru.e:fi MaTeMaTHl!eCKOH TeOpHeii Kpae-
BbIX sa,ZI .all 3JIJIHI1TfftieCKOro, CMeIIIaHHOro H rHnep6oJIHl!eCKOro TH-
noB.
§ 7. napa,11.oKc )J.aJiaMoepa
BoJiee H3BecTHhIM H 6oJiee ,ZI .3BHHM, qeM napaJlOKC oopaTHMOCTH,
HBJIHeTCH napaJlOKC .UaJiaMoepa. CorJiaCHO 3TOMY napa.n:.oKcy, H3
.n : .onyru.eHHH, c ,n;eJiaHHbIX B § 5, CJieJJ.yeT D = L = 0. )lJIH cJiyqaeB
P H c. 1 . 06TeKaHHe . ~HJIHH ~pa.
no 3fiJiepy.
Kpyroaoro UHJIHH,!I .pa (pHC. l) H ccpepbI 3TO CJie,!I .yeT, B CHJIY CHM·
MeTpHH, H3 HBHOH cpopMbI IlOTeHUHaJia CKOpOCTeii:
U = a(x+ :a) (UHJIHHJI.P),
(12a)
U =a (x+ 2; 3 ) (cq:iepa)
(126)
H TeopeMbI BepayJIJIH (8*) npH g = 0. ECJIH sa.l{aqa KOppeKTHO no-
CT8BJleHa, TO HaJIHl!He qeTbipexKpaTHOH CHMMeTpHH, KaK B Jl8HHhIX
CJIYll8HX, Il03BOJIHeT Il0K838Tb, 'ITO D = L = 0. HCXO,!I.H TOJibKO H3
coo6pa>KeHHii o6paTHMocTH (OJ, cTp. 248).
'
Boo6m.e JKe napa;i;oKc )laJiaM6epa cJie.n;yeT Hs npHHUHna o6pa-
THMOCTH Jl.7! .H JII06oro npO(p'HJIH , KOTOpbIH 06Jia,n;aeT tJe1tTpa.1tbHOU
CHMMeTpHeH, T . e . )lJIH TaKoro, KOTOpbIH OT06pa:>KaeTCH B ce6!l npH
1
) Comm. Pure Appl. Math., 9 (1956), 45-68, JO (1957), 107-131 H 11
(1958), 129-144, [CM. raKJKe [4*}, [5*]. - JJpuM. ped.J

28
I' A. I . llapadoKCbl He8'l3KOZO re•teHUJI
OTpa)Kemm OTHOCHTeJlbHO HeilO.D.BH)KHOfO 1~eH1pa CHMMeTplm. 06Te-
KaHHe IlJIOCKOH IlJiaCTHHKH Ha pnc. 2,a .n.aeT npHMep no.n.o6Horo
po.na. llasnemrn, .u eikTBYIOlll.He Ha 9JieMeHTbl IlOBepXHOCTH, COOTBeT-
crnyroru.He Jipyr .n.pyry npH ueHTpaJibHOH CHMMeTpllH, paBHbI no Be-
J1li''lllHe n npo1nBonoJIOJKHb1 no HanpaBJieHnJO. CJie.nosareJibHO, s1a
CHCTeMa CHJI CBO,llllTC5l TOJibKO K nape CllJI ( CM. npHM. 1) Ha crp. 26).
---------
-----------------
Pa3iJe11f/HJ!l/a51r:JI
JI
JIU H/151 !TID/fct
•
---------~~---~~~
-------------
-------
Pa3Den51m1qa51c51
flUII/J51 {llf/hO
PaJ{JenJIHJU{U. ,HC 51
nut1uJ1 moxa
a
6
P fl c. 2 . 015TeKaHHe TIJIOCKOH TIJiaCTflHKH,
no 3:1bepy (a), IlO )l(yKOBCKOMY (6) fl no feJibMfOJihny (8).
lleMOHCTpauirn napa,ll,OKCa a o6ru.eM cJiyqae JI.eJio JI.OBOJlbHO TOH·
KOe, npn 9TOM llCilOJib3Y€TC5l CJIOJKHa51 reopeMa O noBe,lleHHH pellle-
HHH ypaaHeHH5IV 2U = 0 Ha 6eCKOHel!HOCTH. A HMeHHO, nyCTb U(x)
B03pacraer Ha 6ecKOHel!HOCTH, no KpanHen Mepe K3K nepBaH cre-
neHb lxl = r. TorJI.a MO)KHO noKa3aTb, l!TO
U=a · x+<p(x),
rAe qi(x) «perym1pHa Ha 6ecKoHel!HOCTH» ([4], rJI. X, § 8; [2*1). no.u
9THM Mb! IIOHHMaeM TO, l!TO qi(x) MO>KHO pa3JIO)K!ITb B HeKOTOpblH

§ 8. Teopw1 Kpb/./lOBozo npo<pUJIR
29
CXOJf.SHU!ITlCH pn..n: ( aHaJlOrH\!HbIH PH.llY no orpmiaTeJlbHbIM crenemtM
B paaJiomeHHH JiopaHa), tIJI€HhI KoToporo cyn, npo1rnae.nemrn
orp1mareJ1hHhIX creneHeH r H cq>epHqecKHX rapMOHHK, Bb1pa2KeH-
HbIX l!epe3 urnpory H ,[(OJirOTy. (JlmI TaKHX pemeHHH ypaBHeHirn
vzu = O npa».1J .onoA06Ha51 nmote3a (E) noJI..rsepm..n:aercH, CJie..n:o-
BaTeJI&Ho, crporou reopeMoii.)
§ 8. TeopHst Kpbl.' fOBOfO npo<f>HJlst
He cMyIUaHCh np1rne,neHHbIMH Bbime napa.noKcal\rn, ytJ:eHhie cy-
Mem1 npaBHJ1hHO IlOJIYtJJITh, no Kpaiitteti: Mepe K8'4eCTBeHHO, JIOOOBOe
conpOTHBJieHHe H IlO)l'heMHYIO CHJIY, OCT8B3HCb B paMKax ypasHeHHH
.l l.BlUKemrn 3fIJiepa. BcH XrIT!)OCTh 3aKJIIOtrneTCH B TOM, ,no6bI II36e-
)K3Tb ynorpe6JieHirn I'Ffl10T€3hI (D)' KOTOPY!O npHMeHHJIH 3iiJiep H
JiarpaHiK, a 3TO MO)KHO cneJiaTb, HCilOJib3YR pa3pbl8Hbte H .iltt-toeo-
3Hal,f,Hbte IlOT€HilJ1a.iihI. (TaK!fe q>yHKUHH, npaa,Aa, ttacro paccMaTpH·
BaIOTCH «npaKTHKaMH» KaK naToJiormrl)
,[[JI"H 011pe,11;e.Tiemrn 1to6oeoeo conporue1tenua MOmHo nocTyJIHpo-
BaTb HaJIHtJHe aacToti:Hoii KU/tbBarepH.oti 30Hbt { cJieA, o6JrncTb
«MepTBOH BO)].hI») c U = 0 no3aAH rrpemncrBHH, 11pocr11paromeroca
,n:o oecKoHetJHOCTH, KaK Ha pnc. 2, a. 3ra aoHa orneJiena or rJiaB·
ltoro retJ:eHHH «cso6o)l!HbIMH JlHHHHMH TOKa» c IlOCTOHHHblM JJ,aBJie·
HHeM, npulleM CKOPOCTb u = vu H3M€H51€TCH CK8lJK006pa3HO npH
nepexone itepe3 3TH JIHHIIH. 3ra Mo,ne..'Ih 6y.n;er II3YlieHa s § 39.
TeopHIO IlO,A'E>eMHOii CHJ!bl B ,[(ByMepHOM TetieHHH MOiKHO nOJiy·
1:fl1Th, BBO ,ll ,Jl. MHOf03H8t!HbJ'H noreHuHaJI BH)l.a
r6~
U= 2i: + ~akx11 +ip(x),
(13)
V2ip=O H q,(x)=0 (;2) •
3.necb r = ~ dU = f ~ u11 dx1; -11JJPKYJ1HUJHI, orrpe.neJieHnarr s
§ 4, npHtJ:eM HHTerpaJI 6eperc51 BOKpyr npemncrn1rn (npoq>umI
KpbIJia).
BBO.llf! tJJieH nl/2Tt B q>OpMyJiy ( 13), MbI >KePTBVeM neTePMIIHH3·
MOM,u TaK K8K npH STOM 3a,n.aqa H€HMaHa H3 § 4 38MemieTCH 38·
;.i:atieH, KOTOpaH He HBJIHeTC'1 KoppeKTHO IlOCTaBJieHHOH. HeKoropyro
BHAHMOCTb )leTepMHHH3Ma MO)KHO eme coxpamtTh, Kor.na Kpb1JIO
1rneer ocrpyro 33.ll .HIOJO «KpOMKy» ( HO He B o6meM CJiyqae). B 9TOM
CJiyllae RBJIHeTCH ~paB,/l.Ol10,ZJ,06Hl,IM npe.nnoJio>KeHHe, l!TO <. ~C K OpOCTh
KOHetJHa Ha 31:UlHeH KpOMKe» {yCJIOBHe )KyKOBCKOro- 4anJiblfHHa).
3ro ycJIOBHe BW)leJIHeT enffHCrBeHHOe 3Ha4eHne UHPKYJIHUHH r H
TI03BOJIHCT HaXO)lHTb JI060BOe conpoTHBJleHHe H Il0)l'b€MHYIO CHJIY
CJie;.i:yromeii TeopeMe Kyrra - >KyKOBCKOfO ([8J. CTP. 188):

30
I'. n . . I . l1apaiJoKcbt HeBJl3Koeo 1e<teHuJ1.
Teo p e M a 2. B A1060M n.11.oc1CoM reitenuu Buaa ( 13) Mbt uMeeM
D=0HL=par,eaea=la,.
B qaCTHOM cJiyqae r = 0 Mhl nonyqaeM K8K CJieJI.CTBHe napa.U.OKC
,UaJiaM6epa.
CT8UHOHapHoe JIOK8JlbHO 6e3BHxpesoe IlJIOCKOe TeqeHHe c UllP·
KYJISIUHeH MO)KHO onpeJI.eJIHTb K8K «Te'leHJ-Ie )KyKOBCKOfO», ecJIH
OHO YJI.OBJieTBOpsteT ycJIOBHIO )KyKOBCKOfO. Te'leHHe )KyKOBCKOfO
Jl.JISI DJIOCKOH DJlaCTHHKH cxeMaTH'leCKH H306pa)KeHo Ha pHc. 2, 6;
K09q>q>HUHeHT IlO.ZJ. 'beMHOH CHJibI CL = 21t sin a., r.ZI .e a. -
yroJI aTaKH.
Te'leHHe )KyKOBCKoro .ZI .JIR 3aJI.aHHoro npoq>HJISI c ocrpoti 38JI.HeH
KpOMKOH npe.nCTaBJISleT co600 KOppeKTHO DOCTaBJieHHYIO KpaesyIO
aa.naqy. Ee ipemem1e B qacrnhlx CJIY'laHx (npoq>HJib )KyKoBcKoro,
npoq>HJib I<apMaHa - Tpecpcprua H T. .ZJ . .) COCT8BJIHeT OCHOBHYIO
rnaay COBpeMeHHOH TeOp'HH KpbIJia; snepBbie 06m.y10 TeOpHIO ( c
npHJIO)KeHHHMii) .ZI .aJI MH3eC 1). Ee cnpaBe.ZI .JIHBOCTb OCHOBhIBaeTCH
Ha cne.ny10m.e0 reopeMe qffCTOH MaTeMaTHKH, KOTOpaH Il03BOJIHeT
H3M npeo6pa30BhIB8Tb 9JieMeHTapaoe Te'leHHe )KyKOBCKOro (12a)
. ZJ.JUI eJI.HHHqHoro Kpyra B HeC}KiiMaeMOe Te'leHHe )KyKOBCKOro J(JISI
npOH3BOJlbHOf0 npoq>HJrn.
OcHOBHaH TeOpeMa O KOHcpOpMHOM OTO(5pa)Ke-
H H H. H MeeTCJl oona u TO/lbKO oiJna ICOMnllelCCHaJI. aH.allUTU'lecKaJI
</JYHKl{UR
00
w=f (z)= kz+~ckz-k, k>O,
0
OT06pa,«;a10w,aa 63aU.MHO oaH03Hll'lHO u KOH</Jop.MHO o6AaCTb BHe
eiJUHU' -lHOZO ,cpyea Ha BHeWHOCTb OllHHOU oa1WCBJl3llOU o6AaCTU.
B DOCJie.ZI.Hee BpeMH 9TOT pe3yJibT8T 6b!JI pacnpocTpaHeH Ha
«KB83HKOHcpopMHOe» OT06pa:lKeHHe ( CM. npHM. 2) Ha crp. 26)' KO·
Topoe cocTOHT B TOM, tJTO JVIH .ZJ .8HHoro 'lHCJia Maxa M < I HMeeTCH
OJI.HO H TOJlbKO O.ZJ.HO .l(.03BYKOBOe o(heKaHHe, no )KyKOBCKOMy, .l(JISI
JII06oro npoq>HJIH c OCTpOH aa.nHeH KpOMKOH.
B cnyqae xopomo o6TeKaeMhIX npoqmneit npH MaJIOM yrne aTa·
KH .n.eif'CTBHTeJibHhle DOTOKH xopomo annpOKCHMHPYIOTCH H.ZI .eaJib•
HhIMH Te'leHHHMH )KyKoBcKoro. Xon1 noJiaraTh, 'ITO no6osoe conpo·
THBJleHHe paBHO HYJIIO, O'leBH.ll .HO CBepxonTHMHCTHtJHO, TeM He Me·
Hee IlO.U. ' beMHaH CHJia B .ZI.e:ikTBHTeJlbHOCTH COCTaBJIHeT 75-95 %
pacqe'JlHOH, a OTHOllieHHe DOJI. 'beMHa.H CHJia/no6osoe conpOTHBJieHHe
MO:>KeT Jl.OXO.U.HTb Jl.O 50.
1) Mises .R ., Zeits. Flugt. Motorluftschiffahrt, 1917, cTp. 157-163 H 1926,,
CTp. 67-73 H 87-89 . 0THOCHTeJlbHO 8H8JIH38 q>aKTH'IeCKHX .!l8HHblX CM. M ff-
3 e c P., TeopH11 nonera, M., HJI, 1949. (cJ>opMyna 11.nH onpe.L1 .enenH11 MOMeilTa
Cl!Ji, 11.elicTBYIOlllH_X Ha. KpblJIO, 6MJl8 nonyqeHa c. A . 1
- IannbirHHblM (CM. q an JI Ir
r H H c. A. Co11., T. II, M.-JI., 1933). -
llpuM. pea.}

§ 9. 31/>(/Jei.r MtuHgca; aepuBa14uJ1
31
-----------------------·--------
O.zu1a1.;o Y('JIOBHe )Ky1<0BCKOrO HHKOHM o6pa30M He .11aet H3Jle)I{·
noii reopnu no;'lbeMHoii CHJJbl s o6UleM rny 11ael TaK, B rpen1epnoM
npoc-rpaHCTBe o6JJaCTb BHe caMonera, 011eBHllHO, HBJJHeTCH O.!lHO·
CBH3HOH. CJJe}losareJJ&uo, nio6oe noKaJJbHo 6e3BHxpesoe re11en11c B
npocrpaHCTBe JlOJl>KHO HMeTb OJlH03Ha'lHblH noreuuHaJJ CKopocteH
u npH nyJJeBOH 00.ll'beMHOii CHJJe. EcJJH 6bl 9TO 6blJIO n.eflCTBHTeJlb·
HO TaK, noJJer 6blJI 6bJ HeB03MO>KeH.
6oJJee YTOH'leHHblM HBJJHeTCH CJJe.llylOlllHH napa.110KC 4ll30TTII I).
PaCCMOTpHM re11eHHe )KyKOBCKOro .llJIH OJIOCKOii OJJaCTHHKH, cxeMa·
TH'leCKH H306pa>KeHHOe Ha puc. 2, 6. CorJJaCHO reopeMe KyTTa -
)KyKOBCKOro, pe3yJJbTHpyiomaH CHJJa .llOJl)l{Ha 6bJTb HOpMaJibHOH
K DOTOKY; nocKOJJbKY >Ke JlaBJJeHHe BCIOJlY HOpMa.llbHO K nJJaCTHHKe,
3Ta CHJJa .llOJl>KHa 6b1Tb HOpMaJJbHOH K nJJaCTHHKe - 0 11eBIIJlHOe
nporHsope4He. KaK noKa3aJI 4H30TTH, no 061,sicusteTCH coeceM
npocro: Ha 38..llHIOIO KpOMKY ..lleikrsyer KOHe'IH8ff CHJJa BCJie.llCTBHe
6ecKOHe'lHOro orpHUaTeJlbHOro .11aeJ1eHHR (UO..llCOCa). IITO CBH3aHO,
y11HTbJB8H <f>opMyJJy (5). c 6eCKOHelJHblM 3HalJeHHeM CKOpOCTH 8
9TOH TOIIKe. TaKHM o6pa30M, napa.llOKC CBR33H C teM, 4TO iHeCO·
CTOHTeJJbHa rHDOTe3a (E) H3 § I, H MO>KeT 6b1Tb 113383H napa·
JlOKCOM oco6oi1 TO\JKH.
K co)l{a.11eu1110, 9KcnepHMeuraJJbHble .nauuble He nO.!lTBep)l{Jl3IOT
H3MeHeHliff 00.lll>eMHoi'i CHJlbl C H3MeHeHlleM ¢opMb1 KpblJla, yKa::u,1 -
saeMoro reopneii )KyKoBcKoro. Mb! noJ1y11aeM 3JleCb cne.11yiou1nii
napa.110Kc YTOJIUleHHSI: teopeTHtteCKU K09<f><f>HUHellT c,. JlOJJ>KeH 803-
pacrarb c yromueuHeM KpbIJJa; 8 }leHCTBHTeJlbHOCTH >Ke 06blt(HO Oii
y6blsaer 2 ).
§ 9. 3ct,ct,eKT Marayca; AepH88UHSI
HrpoKaM 8 ronb<J> H TeHHHC H3BeCTHO crpeMJleHHe BP3lll8l0Ule·
rocH MRlla YKJJOHHTbCSI OT CBOeii HOpMaJibHOn tpaeKTOPHH B uanpa·
BJieHHH, B KOTOPOM spaUlaeTCSI ero nepe.11.HHSI 48CTb. 310 SIBJ1eH11e
H83bIBaercsi 3<J>cpeKTOM Marnyca. CorJJacHo P3JJeIO ([12), t. I,
343-346), 9cl>cl>eKr Maruyca o6bl'IHO 060Hcnsi1or Ka11ecrseHHO cne-
.nytolUHM o6pa30M.
JloK8JlbH38 CKOPOCTb 803.llyxa OTHOCHTeJlbHO MH'la H3·38 ero
apaIUeHHSI 60JibWe c TOH CTOpOHbl, r.ne BpameHHe uanpaBJleHo Ha·
38.ll , \JeM T3M, r.l(e OHO uanpaBJJeHo Bnepe.11 (CM. pHC. 3). Cne}lOB3·
TeJl&Ho, no ypaaueuuio 6epHyJJJJH (3), .nasneHHe c O.JlHoii cropoHbl
1
) CM. Cisotti G., Rend. Accad. lincei, & (1927), 16-21 11 7 (1928)
17-19 H 538-543; a taK>Ke Plstolesi H., taM >Ke, 12 (1930), 409-411'.
(3tOT napuoKc 6blJJ H3eecteH el.l(e H. E . )KyKOBCKOMy, KOTOpblll .ll8JJ 8 C88311
C nHM 06"bHCHeHHe JIBJleHIIJI DO.llC8Cbl88101.UeA CHJJbl; CM. )I( y KO BC K H ii" H. E.•
O DOJl.llep>KHB8101llHX DJJaHax T1ma AHTyaHeTT, Tpyllbl OTll. (j>lt3. M3TCM . H8}''1H.
0-BaI Jll06HTeJteil ecTeCTB03H8ffHJI, xv. Obin. 2 (1911). -
llpuM. nepe8.)
) Ct.t. (1), crp. 252. -
llpu.u. nepee.

32
I'A. f. flapaaoKCbl He8 !!3K020 Te'leHUJI
MeHblUe, H 3TO .a.aeT pasuo.a .eHCTBYIOlll.YIO B uanpaBJieHHH, COOTBeT-
CTBYIOlll.eM ua6mo.a.aeMoMy.
Ha ocuoaaHHH .o .annoro 061,sicHeHHst otteHb TPY .nHo nonyttHTb
KOJIH'leCTBeHHblH pe3yJibT3T, TaK KaK y Hae HeT KaKoro-JIHOO onpe-
.a .eneHHOro cnoco6a .a .nst Toro, ttT06bl cssi3aTb spalll.eHHe c uupKy-
JISIUHeii -
.a .a>Ke B CJlytiae UHJIHHJl.pa 1). CTpaH.U.TJib npe.a.npHHSIJI re-
pou11ecKy10 nonhITKY onpe.a .eJIHTb xorn 6b1 MaKcnMyM no.a .1>eMuofi
CHJlbl L, KOTOpblH, KaK OH YTBep>KJJ.8JI, JJ.OCTHraeTCSI Tor.a .a . KOr.na
3Ha11euue UHPKYJIHUHH onpe.a .eJisieTCH npu ycJJOBHH, tJTO HMeeTCH
o.a .Ha·eJJ.HHCTBeHHaSI KpHTHlJeCK8Sl TOlJKa 2 ).
OcHOBblBaSICb ua :noM, OH HallleJI, lJTO M8KCHMYM K03cpcpuuueHT8
CL paaeH 41t. He.nasuo 3TO 39aqeu1:1e 6bJJIO npesbl1Ueuo 3) - eme
PH c. 3. 34>4>eKr Marnyca.
OJI.UH ¢aKT, IlOK83blB8i0111.HH HeHa)J.e:lKHOCTb HeCTpornx paccy>K.a .eHHH.
HecocrosireJibHOCTb cy111.ecTBy10lll.HX 061,sicuem-tfl 3cpcpeKra Mar-
uyca elll.e 6onee sipKo noKa3hJBaeT cJie.a .y10lll.HH napa.a .oKc 3cpcpeKra
Maruyca.
CTapaJI.0KC 34>4>eKTa MarHyCa. flpu MQAblX CKOpOCT.RX
Bpatu,eHUR HanpaBA€HUe OTKAOH€HUR 8 0€UCTBUT€AbHOCTU npOTUBO·
noAO~HO TOMY, KOTOpoe oaer o67JRCH€Hlle P:JAeR ( u KOTOpoe HO·
6A10oa1tocb MaeHycoM) •) .
. Unst roro 1uo6bl o6'bHCHHTb 3TOT napa.a .oKc 3cpcpeKTa Marnyca,
Hy>KHO, IlO·BH)lHMOMY, yl!eCTb Typ6yneHTHOCTb norpaJJHlJHOro CJIOH-
1) KaK noKa3aHo B § 8, B cJJy'lae ccpepbl Mbl npHXOJlHM .n .aJKe K 6onee rny·
60KOMY napa.n .oKcy: BCJJe.n .cTBHe 0.D.HOCBSl3HOCTH UHPKYJtllUHSI r = 0.
2 ) }lOBO.D.bl DpaH.D.TJISI H3JIOJKeHbl 8 ([3], § 27); JJ,1SI UHJIHH .n.pa pa.n .Hyca c
npH OTHOCHTeJlbHOii nocrynaTeJJbHOii CKOpOCTH a UHPKYJISIUHSI f = 4 rrca. no no·
80.llY 6oJJee n03JlHHX 3KCnepHMeHTaJlbHblX .a.aHHblX CM. (50], § 239.
3) Swanson W. M., Final Report on Contract DA-33-019-0RD-1434, Case
Inst. Technology, December 31, 1956 . npH V = {t)C = 17 a K03<pcpHuHeHT
CL = 14,7 H «npOJlOJJJKaJJ B03p8CT8Tb c OOCTOSIHHoi! Cl<()pOCTblO>. He.n.aBHO,
GI au er t M. B . (Proc. Roy. Soc. A242 (1957), 108 -115) noJlaepr BblBOJl
IlpaHJlTJlll KpHTHKe, HCXO.LI .SI H3 reopen1 11ecKHX coo6pa>KeHHii.
4
) CM. [3], § 27 H 221; 3KCnepHMeHT3.1bHble .n .aHH!,le (B CJJy'lae ccpepbl) no-
JIY'IHJI MaKKOJI • .llaHHble, DpHBe.n .eHHble 8 (3], § 239, noKaJblBalOT, 'ITO , OO·BH.llll ·
MOMY, 8HaJIOrH'IHasi CHryauHH HMeer Merro H .LI.Jiff UH.111H.Llpa.

§ 9. 3</)tpe1'T .\lacHyca; uepUBUL{UJI
3J
seJieHue, Koropoe .no c11x nop He no.a ..' laerCH Mare:.1arn,1ecKo:.1y
HCCJJe.noeaHHJO KaK KpaesaH 3aJJ.aLJa. TaKHM o6pa3oM, np11 J11060M
KOppeKTHOM HCTOJIKOBaHHII peaJibHOH nonepeLJHOH CH,1bl np11 MaJlblX
cKopocrnx epameHHH na.nJJe)KHT y41nbrnarb 411cJ10 PeHHOJ1b.,rtca 1).
S{sneHHe «.nep11sau1111» auaJ1orHLJHo 3QHpeKry Marnyca. Apr11J1-
nepucraM y:>Ke 6onee cm ner H3Becrno, LJTO spamaJOLUHeCH cnapHJlbl
HMeJOT reH.neHUHJO OTKJIOIIHTbCH OT sepn!KaJlbHOII n.rJOCKOCTII, B KO -
ropoii npoH3BO.!l.HTCH crpeJ1b6a, 11
'! TO ra1we OTKJ10Hem1e npo11cxo-
.llHT B HanpaeJieHHH spameHHH fOJIOBKH CHapH.a .a. OnHaKo 3TO HB-
JieHHe 8 re4eu11e MHOrllX JI er nOHl!MaJIOCb 11enpaBHJlbHO 2 ).
0.llHO HeeepHoe o6bHCHeHHe 6blJIO npeJJ.JIO>KeHO H3BeCTHblM Ma-
TeMaTHKOM - nyaccoHOM. OH ClJIITaJI, '!TO BCJleJJ.CTBHe HHepu1111
OCb CHapH.na orcraer or HanpaeJieHHH KacareJibHoii K rpaeKrop11H,
--------- .......
.,,,- .,,,- -
--
\,v~ /
-
~/
'
er- /
r:11ap_qiJ
'\~IJ./
/
Mr:111,wcc iJm1ne1111,:
u
cs ---- lla'!fl1J(J/IC11ue Dep111mLf.uu
~
6onr,wce iJa1J/lt. 'Jll/1?
6
PH c. 4 . 06bRCHem1e s<ficfieKra Marnyca, no nyaccouy.
KaK cxeMaTHt.J.ecKH noKa3aHo ua p11c. 4, a. C.1enoeare.11bHo, Ha HIDK-
Heit cropoHe JJ.OJI>KHO co3.narbCH 6om,wee .nas .11e1rne, a 3HaLJ11r II
60JibWee rpeuue. B cooreercrnm-1 c pHc. 4, 6 3TO .nom1rno np11secrn
K OTKJIOHeHHJO B ,ffa6J110.laer.10M 11anpaE1J1eHHH. OrnHUOlJHOCTb o6bHC-
HeHHH Tiyaccotta CTaHOBHTCH Q<1eeH;J.HOl1, ec .1111 npHMeHIITb ero K epa-
meHHIO TeHHHCHOro MHLJa: no., yLJHJIOCb 61,1 HanpaBJieHHe OTKJIOHe-
HHH, npoTHBOOOJIO.lKHOe o6bl'IHO:\IY 3cpq:>eKry Marnyca!
npaBHJlbHoe 06bHCHen11e 3aKJIIO'JaerCH s cne.ny1omeM. C no-
MOlllblO KOJIHtteCTBeHHOfO HCCJieJI.OBamrn rnpornonHLJeCKOH ycrOHlJH-
BOCTH MO>KHO yctaHOBHTb, 'ITO ycroHLJHBOe no., o)KeH11e OCH CHapH.:ta
(c npaeoA BHHToeoA ttape3KOH) ttaxo.nnrn cnpaea or KacareJibHOH
K rpaeKtopHH, a He BbIWe ee, KaK 3TO yreep>Kaa.1 nya.ccoH. TaKHM
ot5pasoM• .nepneaum1 cHapsi.na BbI3bJB~errn rnaBHblM o6pa3oM ue-
•) C11. Krahn E., J. Aer. Sci., 23 (1956, 377-378.
I) HatepeCHblii IICTopHqecKHii O(hop ,UaH B (5], fJI. X .

34
f .II. /. llapadOKCbl He6Jl3KOi!O Te'teHUJl
nocpe.ncTBeHHO a3pO.U .HH8MH•leCKOH nonepe•rnoii CHJIOH I{ J!Hillb KOC·
BeHHO - apameHHeM.
3To om1Tb·T8KH noKa3blBaeT HeHa.ne:>KHOCTb Ka4eCTBeHHblX coo6-
pa>KeHHH. BepOS!THOCTb TOro, t!TO cnyqaiiHoe 06'bffCHeHHe OKa>KeTCff
npaBHJibHbIM, paBHa 50% I
§ JO. BoJIHOBOe JI06oeoe conpoTHBJieHHe TOHKHX KpWJibeB
napa.noKc .UanaM6epa HeJib3H pacnpocTpaHHTb Ha caepx3ayKo-
aoe Te4eHHe: .na>Ke 6e3 yqeTa BH3KOCTH MaTeMaTH4eCKHe coo6pa:>Ke-
HHH npHBO.U .SIT K cymecTBOBaHHIO IlOJIO:>KHTeJibHOfO JI06oaoro conpo-
THBJieHHff. BBH .U .Y napa.noKca o6paTHMOCTH 3TO B03MO>KHO TOJlbKO
IlOTOMY, 4TO KpaeBdSI 38.U.848 (JI.Jiff CTaU.HOHapHoro Jl.BH>KeHHH), on-
peJI.eJIHeMaH ypaBHeHHSIMH 3uJiepa, He ffBJISleTCSI KOppeKTHOH. Mbl
flOKa>KeM ceii4ac 3TO, Hat1aB C paccMorpeHHSI JIHHeapH30BaHHOrO
caepX3BYKOBoro Tet1eHHSI (TeOpHH «TOHKOro KpbIJia»).
PaccMOTpHM ceMeHCTBO He3aBHCHMbIX OT speMeHH C>KHMaeMblX
TetleHHH, 38BHCffll.lHX OT napaMeTpa a- TOJilll.HHbl KpbIJia. Mbl npeJI.-
noJiaraeM (nmoTe3a (E) H3 § l), 4TO noTeHuHaJI CKopocni MO)l(HO
38IlHC3Tb 8 BHJI.e
V=ax+acp(x, y. z)+O(a2).
(14)
OoJI.cTaBJIHH ero a q>opMyJiy ( IO) H .neJiasr o6hl4Hbie s TeopHH ao3-
MymenHii .nonymenHH, Mb! nonyqaeM 1) npH a-+ 0
(M2
-l)fxx=<f>yy+<f>zz,
M=a/c.
(14*)
51cno, t1To cnyqasiM JI.03ByKoeoro Tet1eHHH (M < I), 3ByKoeoro
reqeHHH (M = I) H ceepx3eyKoeoro Te4eHHH (M > I) oTBe'laIOT
ypaBHeHHSI B '13CTHblX npOH3BO.U.HbIX COOTBeTcTBeHHO 3JIJIHnTH4e-
CKOfO, napa60JIH'leCKOro H rnnep6oJIH4eCKOro THilOB 2). 3To npoc-
roe 3aMe4aHHe y>Ke yKa3bleaeT Ha ro, '!TO Kpaesasi 3aJI.aqa KOppeKT·
no nocrasnena JIHWb s .no3BYKOBOM cnyqae.
B cJiyqae nJIOCKoro re'!eHHH cp = cp(x, y) ew.e co speMeH .UaJiaM-
6epa H3BeCTHO, t!TO o6mee perneHHe ypaBHeHHSI (14*) HMeer BHJ:l
C¥=F(x-VM2 -1 y)+O(x+ VM2 -1 y),
(15)
rJI.e F(r) H G(s) - npoH3BOJibHble q>yHKUHH.
1) CM. (6), § 141 HJIH (101, CTp. 245 . 60J1ee no.11po6Hoe OflHC3HHe npHJIO>KeHHft
CM. B (10], rJJ. VIII.
2) 3To BepHO T3K>Ke H (5e3 JIHHeapH33UHH, HO B T3KOM CJlytJae M 6y.11eT 33·
BHCeTb OT KOop.nHH3T. CJJe.noBaTeJlbHO, B03MOlKHbl rpaHC3BYK0Bble nOTOKH H CO•
OTBeTCTBYIOIUHe HM .llHCpq>epeHUH3Jlbllble ypaBHeHH!l cMellJ3HHOro rnna (9JIJIHflTH-
qecKHe B 0,ll;HHX o6JI3CTSIX H rHnep60JIHtJecKHe B llPYrHX), K3K fl0K333HO B § 6.
[CMelllaHHblM TetJeHHIIM nOCB!IIUeHa Ot'illlHpH3!1 JIHTeparypa; CM. ,
HanpHMep, [7•]
H [8*}. -
llpuM. ped.J

§ JO. Bo,IHOBOe ,w6oBoe conpOTUBAe,we TOHKUX KpblAbeB
35
.Um1 TOfO 4T06bl onpe.neJIHTb f(r) H G(s), HY>KHO HCnOJib30BaTb
ycJIOBHe (7), KOTOpoe npH CTaUHOHapHOM Tel!eHHH CBO.LlHTCH K pa-
au
seHCTBY cJn = 0 HJIH
:~ = a71' (x)
(15')
.nnsi «TOHKoro KpbIJia», orpaHHl!eHHOro KPHBOH y = TJ (x). Mw 3aMe-
iJ
O
HHJIH B ( 15') an Ha di ' npe.nnoJIO)KHB, tJTO TaHreHC yrJI a H8KJIO-
Ha rt'(x) ¢: 1.
.lleifcTBHTeJibHO, raKaSJ rnnore3a ( HJIH, septtee,
11' (x) ¢: M) HBJISleTCSI OCHOBHblM .nonymeHHeM Teop1rn TOHKOrO
KpbIJia.
lfro6bl H36e>KaTb napa.D .OKCa o6paTHMOCTH H nonyl!HTb KOppeKT·
HO nocTaBJieHHYIO 3a,Lla4y, Heo6XO.ll.HMO CHCTeMy ( 15)' ( 15') .ll.OilOJI·
HHTb HeKOTOpOH .no6aB04HOH rnnore30H Heo6paTUJrtOCTU, Bblpa.>KalO-
meif HHTYHTHBHO Ol!eBH.LlHblH
0
q>H3H4eCKHH cpaKT, 4TO «BOJ!Hbl CK8Tbl·
88IOTCSI BHH3 no retJeHHIO». EcJIH Mb! pacnoJIO)KHM TOHKOe KpblJIO
B.LlOJib OCH X, TO nocne.nHIOIO rnnore3y MO>KHO 3anHCaTb B cne.ny10·
meM eH.ne:
IF(x- VM2-1 y), ecnH y <0,
C?(x,·y)=
O(x+ VM2-1 y), ecJJH y >0.
(15*)
C yt1eroM pe3yn&rara no.ncraHOBKH B ypaeHeHHe 5epHynnH (5)
Hawa CHCTeMa ypaBHeHHH n03BOJIHeT 38KJIIOtJHTb O cymecTBOBaHHH
BOAH.OBOZO OGBAeH.UJl, KOTOpoe, B npH6JIH>KeHHH reopHH B03Myme-
HHH, nonyqaeTcsi s BH.Lle p = pa2TJ' (x) Ha eepxHeif nosepxHocrn
KPblJia Y = TJ (x) H 8 BH.Lle p = pa2~ ' (x) Ha HH>KHeif nosepxHOCTH
Kpblna y = ·~ (x). Onpe.neJIHB npo.noJibHYIO cocraeJJsi10my10 .nasne-
lffHH a BblOOJIHHB HHTerpapoeaHHe, Mbl nonyl!HM .LlJISI no6osoro
conpOTHBJieHHSI D = f p dy Bblpa>KeHHe
D=pa
2
f(11'd11+ ~' d~) =pa2fh'
2
+~'
2
] dx,
(16)
r.ne HHrerpan 6epercSJ no .nmrne KpbIJia .
. llnH .LlOCTaTOllHO M8JlblX yrnos HaKJIOHa npHse.neHHble cpopMyJlbl
snonHe xopowo cornacy10rcsi c 3KcnepHMeHTOM ([IO), crp. 346, 350)
H, OlleBH.LlHO, .Ll810T nOJIO>KHTeJibHOe csepX3BYKOBOe «BOJIHOBOe JI06o-
BOe conpOTHBJieHHe». Jl1060nbITHO,
'ITO
OHH cornacylOTCSI c Ol!eHb
crapoA KBa3H3MnHpHllecKoA cpopMynoif 3ifnepa, s Koropy!O sxo.nHT
YHHsepcanbHblH nocrosiHHblii MHO>KHTeJib, onpe.nensieMblH, no npe.n-
nono>KeHHIO, 3KCnepHMeHT8JlbHO I).
1
) 0rHOCHTe.1bHO np1rneHeHHll K 6a.1 .111crn•1eCKII/.I 3a;J.alJaM C~J . (5], § 12-16.

36
f A. I. llapadOICCbl He8R31C020 re'leHUJI
§ 11. TOHKHe TeJla apaw.eHHSI
CnHWKOM CJIO>KHO paccMarpHBaTb 3.lleCb npHMeHeHHe ypaeHeHHH
( 14*) npaHJJ.TJISI - fJ1ay3pra K ceepxaeyKOBOMy o6TeKaHHIO TaK
Ha3blB8eMblX «TOHKHX>, HJIH «YJJ.JIHHeHHblX>, TeJI npOH3BOJlbHOH
<f>opMbl I). Mbl TOJlbKO npHBeJJ.eM HeCKOJibKO npHMepos, HJIJIIOCTPH·
PYIOlllHX 06lllHH Te3HC O TOM, lJTO ecJIH peayJlbTaTbl He nony11eHbl
MaTeMaTHlfeCKH H <l>H3HtJeCKH CTporo, TO HM npncyma TeH)leHUHSI
CTaHOBHTbCH HeHaJJ.e>KHblMH.
OrnocnreJibHo npoc1y10 aaJJ.a11y npencraBJIHeT co6oii oceeoe 06-
reKaHne reepJJ.blX reJJ epameHHSI ( aprHJIJiepniicKne cHapHJJ.bl 6e1
pb1CK8HHSI}. KapMaH H Myp 2) nepeblMH npHWJlH K BblBO.lly, lJTO Ha-
JIHtJHe BOJIHoaoro no6osoro conporHBJieHHH Bb13bleaer peaKHH pocr
conpOTHBJieHHSI npn .llBH>KeHHH TOHKOro CHapH.!18, KOrl{a M = I, H
01.{eHHJIH 3TO B03paCT8HHe conpOTHBJieHHH Ha OCHOBe ynpomeHHH,
yKaJaHHblX B § IO. Bonee 11eM tJepea IO ner Konan pacnpocrpaHHJI
3TOT BblBO.ll Ha cuapHJJ.bl C pb1CK8HHeM H DOK838JI, lJTO ynpomeHH8SI
reopusi npHBOJJ.HT K pHllY own6otJHblX aaKJ110tJeHHH 3). B tJaCTHOCTH,
a cny'lae KOHycoa no.zt yrnoM araKH nonepetJHaH cnna, no.ztc11nraH-
uasi no <t>opMyJiaM HJ § 10, y61,1eae1 c eoapacraHHeM M, e ro epe-
MH KaK npaBHJlbHOe npH6JIH>KeHHe no reopHH B03MyllleHHH )laer ee
yeeAu'l.eHue (napa.ztoKc Konana).
8 HaCTOSllllee speMH npH3HaHO (CM. npHM. l) Ha 3TOH crp.),
1110 npocraR JIHHeapnaoeaHHaR reopnR, npHeeJJ.eHHaR a § 10, JJ.a>Ke
JJ.JISI TOHKHX TeJI npHBOJl.HT K HenpaeHJlbHOMY 3Ha11eHHIO CHJibl. 8 cny-
tJae o6reKa.HHSI ceepxaeyKOBblM DOTOKOM TOHKHX TeJI epameHHSI,
xea.11parHtJHble 11neHbl B ypaaHeHHH BepHynnn npu nOJJ.CtJere JJ.aBJie-
HHR 6y.11yr roro >Ke nopRJJ.Ka eeJIHlfHHbl, 1110 H JJHHeAHblA 11JJeH 4).
11.JHI HeKoropblX lf8CTHblX npHJJOJKeHHA npOCTble JIHHeapH3088H·
Hble ypaBHeHHH H3 § IO HY>KHO BHJJ.OH3MeHHTb TeM HJIH HHblM cno-
co6oM 5). TaK, JJ.JIH KpblJibeB KOHe11Horo paaMaxa no.11 yrnoM araKH
ny>KHO p,accMarpnaarb c6era10mne eHxpeeble CJIOH. KpoMe roro,
1) CM. Ward G. N., Linearized theory of steady high-speed flow, Cam-
t>rid'e Univ. Press, 1955.
) Trans. Am. Soc. Mech. Eng., 54 (1932), 303-310 . 0THOCHTI!JlbHO coape-
MeHHoA <JtHHeapHlosaHHoA> reopHH CM. [IC], § 8.3; HJIH [15), rJt. VIII, § 15.
1) Ko pa I Z., Phys. Reri .• 71 ( 1947), 474; [JO], crp. 377; nonpo6Hee -Tables
of supersonic flow around yawing cones, Mass. Inst. Technology (1947),
ocot5eHHO cTp. XVI, XVII. 3KcnepHMeHTaJibHWe JlBHHWe npHBOJUIT Ho It M. H
BI a ck I e J., I. Aer. Sci., 23 (1956), 931-936.
•) no-BHJlHMOMY, anepawe !ITO noKalaJI light hi 11 M. J ., Reps. Mem.
Aer. Res. Comm .•
2003 (1945); CM. raK>Ke Broder I ck J. B ., QJMAM, 2
(1949), 98-120 H [10], CTp. 307.
6 ) Ursell F. and Ward G. N., QJMAM, 3 (1950), 326-348; Gold-
stein S.,
Proc. Int. Math. Congress, Cambridge, 1950, T. 2. oco6eHHO
crp. 288-289; Ad ams M. C., Se a rs W. R.. /. Aer. Sci., 20 (1953), 85-98.

§ 12. llapadOKC 3pHtuOy
37
e CJiyirne 3aKpyrneHHblX Ten epameHHH, T .aKHX KaK c<pepa, JIHHea-
pH30BaHHaH KpaesaH 3a.l{a11a, onpeJ1.eJIS1eMaS1 nocpe.ncrnoM ypas-
HeHHH ( 14*) u ( 15'), JI.aeT HecymecTBylOlllHe oco6e~HOCTH s KpHrn:
11eCKHX TO'- IKaX (T. e. Ha OCH CHMMeTpHH). Ho caMbIH cymecTBeHHbIH
.u .e<t>eKT reopHH «TOHKOro KpbIJia» 3aKJIIO' - laeTCSl B TOM, 4TO OHa He
8 COCTOSIHHH npeJI.CKa33Tb cymecTBOB3HHe yaapH.blX BOAH..
YJJ.apHble BOJIHbl JierKO Ha6JI10.1J .alOTCSl 8 BHJI.e 11eTKHX JIHHHH Ha
MrHOBeHHblX <j>ororpaqnrnx .l{BH}KeHHSl CHapH.lI.OB, TaKHX, KaK CHH-
MOK, H306pa}KeHHblH Ha <ppOHTHCilHCe. B cny11ae KOHYCOB H JI.pyrnx
ocrpOKOHellHblX TeJI npu JJ.OCTaTO' -I HO
60JibllIHX 4HCJiaX Maxa 3TH
BOJ!Hbl «npucoeJI.HHeHbl» K eepllIHHe flOJI.06HO xapaKrepHCTHK3M pe-
weHHH JIHHeHHblX rnnep6oJIH4eCKHX JI.HcpcpepeHUH3JlbHblX ypasH·e-
HHH. 8 JI.pyrux }Ke cny11affx OHH «OTXOJI.HT» OT eeplllHHbl H OK33bl-
83IOTCSI npH 3TOM BnepeJI.H CHapHJI.a - TaM, rJI.e no JIHHeapH30BaH-
HOH TeopHH He .II .OJl}KHO 6blTb HHKaKoro B03Mylllemrn.
§ 12. napa.11 .oKc .3purnoy
OoHHTHe «yJI.apHoii BOJ!Hbl» MO}KHO T3K)Ke BbIBeCTH Teoperuqe-
CKH, ornpaBJIHHCb OT npOCTOro napaJI.OKCa, KOTOpblM Mb! o6H3aHbl
3pHwoy 1). Halli opraH cnyxa CBHJI.eTeJibCTByeT, 11To asyK npoxo-
JJ.HT 60JibWHe pacCTOSIHHH Il04TH 6e3 HCKa)KeHHH H c noCTOSIHHOH
CKOpOCTblO, 33BHCHuteii OT TeMnepaTypbl 803.lI.yXa. 3TOT OilblTHblli
ql3KT JI.eJiaeT npaBJI.OilOJI.06HbIM npeJI.nOJIO)KeHHe, 4TO IlJIOCKHe asy-
KOBble BOJ!Hbl pacnpocTpaHHIOTCSI B HJI.eaJibHOM HeBH3KOM raae, He
HCK3}K3HCb H He aaryxaH. 0JI.HaKO 3TO He TaK, 11TO IlOK33bIBaeT na-
pa.l{OKC 3pHlllOy.
O a p a .II. 0 KC 3 p H ill O y. fl pU a8ua6aTU'leCKUX KOAe6aH.UJlX
ea3a llAOCKUe 3BYK08bte crau,uoH.apH.bte (JO,lH.bl KOH.e'lH.OU QMlllltlTy8bt
MareM.aTU'leCKU H.e803J,!OJICH .bl .
. I 1 . o Ka a a re JI b c Tso. Ope.nnoJIO)KHM, 11ro cpopMa 3BYKOBbIX
BOJIH HeH3MeHHa H 4TO BOJlHbl pacnpocTpaH5IIOTCSI C nOCT051HHOH
CKOpOCTbIO, HOpMaJibHOH K BOJIHOBOMY qipoHTy. Tor.na, eCJIH Mb! ne-
peAJJ.eM K OCSIM KOOp.lI.HHaT,
.lI.BH)KYlUHMCSl BMeCTe c BOJJHaMH, TO
YBH.!J.HM ,
'-ITO
.lI.BH)KeHue )KH.IJ.KOCTH He TOJlbKO 0,LlHOMepHO, 110 H cra-
U,UOH .apH.o . Bb16pae B Ka11ecrne HanpaBJJeHHSl .UBH)KeHHSl OCb X, Mb!
MO>KeM HanHC3Tb p = p(x), u = u(x) H T.
,'!:.,
H (6e3 yLJeTa CHJlbl
UDKecrn) ypasHem1e BepHyJJJJH (8) cse,a:eTCH K BH.UY udu + dp/p = 0.
I<poMe roro, ypasueu11e Hepaapbrn11ocrn ( l) nepeiiJler s paseHCTBO
1) Earnshaw S .. Phil. Trans .. 150 (1860), 133-148· CM. raK>Ke Sto-
~7e s, Phil. Mag., 33 (1848). 349; Rank i n e \V. J. M.. Phil. 'rrans., 160 (1870),
[61
7: (12], T. 5, 573 (11,111 Proc. Roy. Soc., A84 ( 1910\ 274-284); [7], § 283;
• §51.

38
I'A. /. flapaaoKCbl HeBJl3K020 Te'leHUJI
pu = const = C, HJIH u = C/p. no.ncTaBJIHH 3TO s npe.nb1.nyw.ee co -
OTHOllleHtte, noJiy4aeM ypaBHeHHe
c2 dp
dp
c2 dp
-- -- + -=O. HM{ dp=-2 -·
(17)
p3
p
p
CJie.!lOBateJibHO, no ,!l06Hoe BOJIHOBOe .ZJ .BH)KeHHe B03MO)KHO TOJib-
KO B cJiy4ae, eCJill )Kll.ZJ .KOCTb y.ZI.OBJieTBOpHeT ypaBHeHHIO COCTOHHHH
(3) 4acrnoro su.na:
c2
P=Po--·
p
(18)
Ho HaM He H3BecreH HH O.!lHH ra3, . ZJ .JIH Kornporo a.nua6arn4e-
cKoe 1) ypaBHeHHe COCTOHIIHH HMeJIO 6bI TaKOH Bll.ZJ . .
§ 13. 803HHKHOBeHHe y.napHOH BOJIHbl
KaK H MHorne .!lpyrne napa.noKCbI, napa.noKc 3pmnoy co.nep)KHT
B ce6e 3epHO cymecrneHHOH HCTHl!bl. Dpn 6oJiee Tlll.aTeJibHOM HC·
cne.noBaHHll COOTBeTCTBYIOlll.HX ypaBHeHHH MO)KHO yctaHOBHTb,
'-!TO
.Z I.JIH a.nua6aTH4eCKOro TetteHHH ra3a 6oJiee nJIOTllbie 43CTH BOJIHbI
KOHe4HOH aMnJIHTY.ZI .bl HaroHHIOT Mettee nJIOTHbie ll B KOHe'- IHOM C' -le-
te neperoHHIOT HX. DoKa3bJBaetCH 3TO CJie,!lyIOlll.llM o6pa30M.
nycTb a 060311a4aer BCIO Macey )Kll,!lKOCTH CJieBa OT .naHHOH
f04Kll, TaK 4TO x(a, t) npe.!lCTaBJIHet co6on nOJIO)KeHHe 43CTHUbl a
t
ax
u
6
1T
B MOMeHT BpeMeHH , a -:;---- eCTb y JleJibHblH O 'beM a = - , or.na
va
p
D/Dt H3 § 3 3aMeHHeTCH Ha (iJ/iJt)a, (ox/ot)a = u ll (o 2x/ot2 )a ecT'b
cy6cTam.1.ttoHaJib11oe ycKope11tte; 3.necb HHlleKc a 03Ha4aer, 4TO Be-
JIH' -IH Ha a ocraetCH nOCTOHHHOH. YpaBHeHHe 11epa3pb1BHOCTH ( l ')
YJJ.OBJieTBOpHeTCH aBTOM3TH4eCKH, noCKOJibKY Dp/Dt = -p2iJ2x/ot oa
ll div U = po2x/oa at. KpoMe Toro, MO)KHO HcnOJib30B3Tb ypaBHeHHe
COCTOHHHH (3) • .ZI.JIH TOro 4T06bI HC.KJIIO' - ll lTb p nocpellCTBOM COOT·
HOllleHHSI
h-1 (I) H
ax
P=
a = (a)=Ha;;,
(19)
Cne.n.oBateJibHO, ecJIH 11e Y4HTbIBaTb cttJiy TH)KeCTH, ·T O q,opMyJibI
:cl)- (3) 3KBHBaJieHTHbl ypaBHeHHIO
(20)
1) ,llenCTBHTC.%HO, ecnu npttHHTb (18), TO noJJy'lHM d2p/dp 2 < 0, qro npoTH-
eopequr eropoMy Ha1Ja11y repMOAnHaMHKH,

§ /4. TepMoauHQ,IIUKQ HeBJl3KUX JICuaKocre,1
39
CTyaCCOH OTKpblJI B8>KHblll KJI8CC peUJellllH ypa1rneHHH (20), 38·
.. l laBaeMblii ¢opMyJioii
u=O(x-(c+u)t).
(20•)
r..lle G(r) - npoH3BOJibH3H cfJYHKUHH. 3rn peweHHH 6b1JIH Ha3ea"!>'
npOCTblMU BOJIH3MH ([6]. CTp. 92); OHH xapaKTeplBYIOTCH CBOH-
a
CTBOM: u = f c(o)do/o, r.LI:e c2
=dp/dp=-o
2
dp/do ecTb KBa.L1 .paT
CKOpOCTH 3By~a. KOTOpaH HBJIHeTCH cfJYHKUHeH y.Ll.eJJbHOfO o6beMa 0:
KaK noKa3aJI AJtaMap, JII06aH nJJOCKaH eoJJHa, TOJJbKO c O.Ll.HOH
CTOpOHl,I BCTyna10maH B >KH.Ll.KYIO cpe.Ll.y ( 06b14HYIO )KH,LI .KOCTb HJJH
ra3), B H8 '18JlbHblH MOMeHT H3XOJlHlUYIOC51 B COCTOHHHH noKOH, )lOJI·
>KHa 6b!Tb npoctoii BOJIHOH.
TaK KaK u=Jc (a) do/a, TO cfJopMyJJa (20*) aaeT cf>YHKUHOH3Jlb-
HOe COOTHOWeHHe Me>K.AY u H a, a CJJe,Ll.oBaTeJJbHO, H Me)K.Ll.Y u H c.
. lle.naH CHOBa nOACT3HOBKY B (20*). npH p = kp r (i > I). JJ~fKO no-
K333Tb, 'ITO (So.nee nJIOTHble '13CTH ra3a H aroHHIOT MeHee nJIOTHbie
((6), CTp. 96). npu11eM c noCTOHHHOH CKOpOCTblO. CJJe.Ll.OB3TeJJbHO,
B Te'leHHe KOHe'IHOrO npoMe>KyTKa epeMeHH tteH36e>KHO B03HHKaeT
pa3pbl8 nAOTHOCTU HJIH «yAapHaH BOJJHa>, lJTO HaXO, . '. lHTCH B caMOM
ffBHOM npoTHBOpe'IHH C runoTe30H ( E) H3 § I.
§ 14. TepMO.l{HH8MHK8 HeBSl3KHX >KHAKOCTeH
.l lm1 Toro 'IT06bJ o6bHCHHTb HBJieHHe yaapHOH BOJ!Hbl, He06XO,LI .H·
MO npHBJie'lb HeKOTOpble Ba>KHble TepMO,LI.HHaMH4eCKHe OOHHTHH 1);
0..llHHX MexaHH'leCKHX KOHI..J .em .1 .Hii ,LI .JIH 3TOro He.Ll.OCT3T04HO . TaK, Ha-
npHMep, Heol5xO)lHMO paccMaTpHB3Tb BHyrpeHHI0/0 :mepcUIO :>KH.Ll.-
KOCTH £ (p, T) )la>Ke TOf)la, KOfJl3 ee MO>KIIO HCKJIIO'IHTb H3 OKOH-
lf8TeJibffblX ypaBHeHHH, KaK B CJJy•iae aJI.Ha6aTH4eCKOfO Te'leHHH.
3-ra
·eeJIH'IHHa BXO)lHT B 33KOH coxpa11eHHH 3HeprnH COfJ13CHO cf>op-
MyJie
dQ=dE+pdV.
(21)
3)leCb p dV ecTb JlHq>q>epem.1.HaJJ pa60Tbl npH OTCYTCTBHH BHeWHHX
CHJI.
Cosepw.eHHbltl za3 MO>KHo onpe,aeJJHTb nocpeacrnoM ypaetteHHH
9A.nepa, TepMOJlHHaMH '!ecKoro ypastteHHH cocrnHHHH p = pRT, rJle
1) TipeBOCXOJlHOe H3.10iKeHHe rept!O.llHHaMHKll <:nO!MaeMUX lKH.llKOCTeii CM.
B [IOJIUIHJIHnMaHr.B.,
p OIllKO A.,
9JJeMeHTbl ra30t!OH JlHHaMIIKH, M. ,
Hn, 1960. (Ha pyccKOM 1!3.: 3 e ., b .ll o s 11" 5l 6., Teoptt11 yaaptt1,1x so.11n
H BBelleHHe a ra3on11ttaM11~y. M. , H 3 .ll. AH CCCP, 1946; 3 a y 3 p P. , Bee,rteH11e
B ra3osy10 AHH3MHKY, M. , r oc r exHJ.!lar, 1947.
-
llpu,11 . :iepeB.)

40
I'It. f. fIapaiJOKCbl He8Jl31Wc0 TetteHU!l
R - ra3oBaH nocTOHHHaH, H cpopMyJihI )lJIH BHyTpeHHeH ~rnepr1rn: E =
= CvT, rne Cv - em.e o,nHa rrocTOHHlJaH (y.ueJibHaH rerrJioeMKOCTh
npH IlOCTOHHHOM o6beMe).
, lI.JIH U30TepMUtteCIWW TetieHHH T = const H H3 COOTHOIII€HH5i
p = pRT cJie.nyeT ¢opMyJia (3a) nptt i = I. B cJiy1iae aoua6arutte·
CKOcO TetieHHH npe)lilOJiaraeTCH, tITO TeilJIOTa nepeHOCHTCH TOJ!bKO
nocpe)lCTBOM KOHB€K[(HH (HeT HH TenJionpoBO)lHOCTH, HH H3Jiyqe-
HHH); npH 9TOM HMeeM dQ = 0 B <popMyJie (21). , lI.JIH e)lHHH<IHOH
Maccb1 (TaK tJTO V=f)HMeeM TOr)lapV=RTH.E =CvT=
= (Cv/R)pV. Tor,ll ,a cpopMyJia (21) )laeT B pe3yJibTaTe ypastteHHe
O=dE+pdV=(Cv/R) Vdp+(1 +Cv/R)pdV.
DoJiaraH i = (R + Cv) /Cv, rroJiyqaeM coornomemre dp/p =
= -1dVJV = 1dp/p, H3 KOToporo cJie)lyeT «nomnporrnoe» ypasHe-
HHe COCT05lHH5l (3a) : p = kp'V.
Co6eputeru-ty10 JfCUaKOCTb MOJKHO onpe)leJIHTb nocpe)lCTBOM ypaB-
tfleHHH 3HJiepa H ycJIOBHH HeOKHMaeMocrn V = const (ypasHe·
HHe (36)).
YpaBHeHH51p9HKHHa- fIOrOHHO.
HcnoJih3YH 3aKOHbI coxpaHemrn Macchi, KOJIHtJ:eCTBa )lBHiKeHirn
H 9HeprHH, MO)KHO TaK)Ke HaHTH COOTHOllleHHe M€)K)ly 3HatieHHSIMH
. naBJieHHH, nJIOTHOCTH H TeMnepaTypbI Pt, Pt, T1 nepe,n y,n:apttoiI BOJI·
HOH H suatJ:eHHHMH Tex )Ke BeJIHtJ:HH Pz, p2, T2 sa y,AapHOH BOJIHOH.
HanpHMep, )lJI5I COBepmeHHOfO ra3a 9TH BeJIHtIHHbl 33BHC5IT TOJ!bKO
OT O)lHOro napaMeTpa - OTHoIIIemrn .naBJieHHH P = P2IP1 HJIH Ull·
TerlCU6H.OCTU CKGttKa p - I . Tor.u2 IIOJIYtIHM ([10], CTp. 30) CJie-
JlYIOllJ.He paBeHCTBa:
P2_ P+~
ri"- ~P+l'
1
_
[t + 6+ l)(P-1) ]2
C-C1
.
2"(
•
DapaJlOKC o6paTHMocrn:, B CHJIY KOToporo MOiKHO 61,rJio Ohl
IlOMeHHTb MeCTaMH HH)leKCbl 1 H 2 B npe)lII1ecrnyrouu1x cpopMyJiaX H
npHH5ITb P < l, MO)KHO ns6e}KaTb, eCJIH npHBJietib BTOpoe HatI,:IJIO
TepMO)lHH3MHKH. (B § 13 npHHIJJIIT, COrJiaCHO KOtopoMy 6oJiee IlJIOT·
Hble tJ:aCTH 6apoTpOmlblX T€tieHHH HarOH5!10T Mettee CTJIOTHbie, npn-
BO)lJiT K TaKoMy }Ke saKJIIoqettnro. Eho CJie,nyeT H3 nepaBeHCTBa
i = (R + Cv)/Cv > l, KOTopoe B CBOIO oqepeAh CJienyeT H3 nuJio-
:iKHTeJibHOCTH BeJIHtJ:HH R H Cv B cnJiy q,n3nqecKHX cooopa.1Kettnfl.)
CooTHOllleHHH AJIH KOCblX y.uapHbIX BOJIH MOiKHO JierKo BblBeCTH
H3 COOTHOIIleHHH .UJI5:! HOpMaJlbHblX CKatIKOB YIIJIOTH€HH5!, HCilOJih·
syH IlO,ll;BH>KHble OCH (§' 67),

§ /5. 5ypyHbl U 6opbt
41
§ 15. 6ypyHbl H 6opb1
Me)KJI,Y JJ.JIHHHblMH 6e3BUXpeBblMU rpasHrauHOHHblMH BOJIHaMH
B )KHJJ.KOCTH IlOCTOHHHOH MaJIOH rny6HHbl H BOJIH3l\lH C)KaTHH B
aJJ.Ha6arnqecKOM ra3e npH l = 2 cymecrsyer 3aMeqareJibHaH aHa-
JIOrHH . .[J.JIHHHbie rpaBHTaUHOHHhle BOJIHbl 6ecKOHeqHo MaJIO~Mn-
JIHTYJJ.hl pacnpocrpaHHIOTCH c IlOCTOHHHOH CKOpOCTblO c = -v gh 6e3
H3MeHeHHH CBOeH cpopMbl, COBCeM KaK npH JIHHeapH3083HHOM npu-
6JIH)KeHHH CBepX3BYKOBoro reqeHHH B § 10 . .[J.JIHHHbie rpaBHT3UHOH-
Hble BOJIHbl ICOHettHOU aMnJIHTY.ll.bl pacnpocrpaHHIOTCH co CKOpOCTblO
JI gh, KOropaH B03pacraer C yseJIHqeHHeM MeCTHOH BblCOTbl BOJIHbl.
CJieJJ.osareJibHO, rpe6eHh BCHKOH JJ.JIHHHOH soJIHhI Ha MeJIKOBOJJ.be
HaroHHeT Bil8JJ.HHY raK, KaK 3TO onHCaHO B § 13. HaKJIOH cppOHT8
BOJIHbl nocreneHHO craHOBHTCH BCe Kpyqe, noKa OH He CTaHeT Bep-
THK8JlbHblM, H BOJIHa, HaKOHeU, «o6pyurneaerCH» no.II. co6crseHHOH
TH.>KeCTblO.
P3Jieii 1) ucnoJib30BaJI 3ry aHaJiornIO, qro6b1 KaqecrseHHO
06"bHCHHTb npeepameHHe B «6opbl» npHJIHBHblX BOJIH rrpu HX pac-
npocrpaHeHHH B YCTbHX peK. IlOJJ.06Hhle «6opbl» noJiyqalOTCH 4aw.e
ecero B nocreneHHo cy.>KaIOmHxcH ycrhHX co cryrrewiarhIM .II .HOM:
OTHOCHTeJihHaH BblCOTa npHJIHBHblX BOJIH yeeJIHq~rnaercH BCJie.II . -
CTBHe noJiyqaIOmeiicH KOHU.eHrpauuu BCeH 3HeprnH BOJIHbl B MeHb-
weM nonepeqttoM ceqeHHH H Ha MeHbllleH JJ.JIHHe BOJIHbl [paBHOH
npoH3BeJJ.eHHIO (12 qacoe) JI gh].
Hoeoro MareMaruqecKoro ycnexa yJJ.aJiocb JJ.06HThCH 6JiaronapH
3aMetJaHHIO P.sr6yWHHCKOro (cM. npuM. 1) Ha 3TOH crp.) , KOTOpbIH
yKa3aJI, qro <PopMyJia c = V g (h + y), rJJ.e y - JioKaJibHaH Bhicora
BOJIHbl, COOTBeTCTByer Bbl6opy l = 2 e COOTHOWeHHH (3a). BcKope
noCJie 9Toro .[{>1<ecpcppu (cM. npuM. l) Ha 3TOH crp.) npHMeHHJI
HJJ.e!O, aHaJIOfHqffylO HJJ.ee P3JieH npu HCCJieJJ.OBaHHH pa3pyllleHHH
BOJIH Ha OTJIOrux OTMeJIHX. no Mepe Toro KaK BOJIHbl rrepeXO.ll.5IT
Ha MeJIKOBOJJ.be, HX CKOpOCTb yMeHbWaercH. 113-33 3TOro 3HeprnH
BOJIHhl cocpe.n.oroqueaercH Ha 6oJiee KoporKoM yqacrKe, qro ew.e
6oJihwe YBeJIHqueaer Bblcory BOJIHhI u ee KPYTH3Hy. EcJIH orMeJib
.ll.OCraroqHo noJiora, rpe6eHb BOJIHbl CHOBa nona.II .aer so ena.n .HHY,
o6pa3yH «6ypyH> npu6oH.
CroKep 2) H .ll.pyrue aeropbl IlblTaJIHCb o6"bHCHHTb KOJIHqecrBeHHO
06pa3oeaaue «6ypyHOB> H «6opoe» npu noMomu ewwenpuse.II .eH-
1
) Proc.
~ oy. Soc. , A9 0 (1914), 324-328; [7]. CTp. 175 -177, 182; CM. raK:iKe
Ria b ouch ins k ·y D., Comptes R~ndus, 195 ( 1932), 988-989; [6], crp. 32 -35 .
PeayJihrar1,1
.l l.>KecpcppH CM. Corn Is h V. ,
Ocean Waves Cambridge 1934
CTp. 154 -)59.
•
•
•
2
) CToKep .ll.>K., BonHbl Ha BOAe, M., l1JI.1 1959, rJI. 10, § 7, § 10 H npH-
ae.11.eHHaJI raM JIHreparypa,

42
L,. /. l1 apadol<:Cbl HeBJl3KOi?O reiteHIIJI
HblX coo6pa)KeHHH. 3TO 3H84HT , l!TO OHH IlblT8JIHCb paccMaTpHB8Tb
3TH SIBJieHHSI B paMKax pauHOH8JlbHOH rHJlPOJlHH8MHKH JlarpaH:>Ka.
0.11 .HaKO npe.11 .cTaBJIHeTCSI COMHHTeJibHblM, l!TO .U .BH>KeHHe )KH.!I .KOCTH
B .neHCTBHTeJibHOM npe6oe H B npHJI'HBHblX BOJIHax HBJIHeTCSI 6e3-
eexpeeblM H8CTOJlbKO, 4T06bl TaKaH MO)leJib 6blJia peaJIHCTH'IHOH.
B HaCTOHllleM npe6oe H B H8CTOHlllHX npHJIHBHblX BOJIHax ecer.na
HMeeTCH 3H84HTeJibH8H 38BHxpeHHOCTb H3-3a OTK8Tbl88HHH npe.11 .-
wec1ey10mex BOJIH ( «nOJlMbIB»), H3-38 Tel!eHHH eceii MaCCbl
>KHJI.KOCTH H r. Jl. H , B03MO)KH0, 83-38 «paccJioeHHH» (CTpa1n<l>HK8·
UHH), Bbl3blBaeMoro eaJIHl!HeM B3BeWeHHOro necKa. BcJie,nCTBHe
3TOro peaJibHble 6ypyHbl MoryT «HblpHTb», «nepeK8TblB8TbCH» HJIH
«pacnJieCKHBat&cH», a peaJibHble 6opb1 MoryT npo,nsHraTbC.SI
.1 en.ne
H30JIHpOB8HHOH CTeHbl BOJlbl HJIH B BHJle CTyneHeK 1). Ka:>KeTC.SI Ma-
JIOBepo.SITHblM, l!T06bl 6e3BHXpeshle rpaBHT8UHOHHble BOJIHbl ,naeaJIH
T8KOe pa3HOo6pa3He HBJieHHH.
KpoMe Toro, CJie.nyeT scnoMHHTb,
·l!TO a a6cTpaKTHYIO reope10
BXOJlHT aBa napaMerpa: OTHOWeHHe hf').. rny6HHbl K JI.JIHHe BOJIHbl
H OTHOWeHHe h/R rny6HHbl K MHHHM8JlbHOMy pa,neycy KpHBH3Hbl
nosepxttocrn R. KaK noKa3aJI e 1925 r. CrpoiiK 2), npe JIJ06blx <fJu,c-
cupoBaHHbtX h H ')... BOJIHbl Jl0CT8TO'IHO MaJIOH KOHel!HOH aMUJIHTYJlbl
MoryT pacnpocTpaHHTbCH 6e3 H3MeHeHHSI ceoeii 4>op,Mb1; 3TO BHJlH-
Moe npoTHBOpe4He c BbJBOJ!.8MH P3JieH H P.SI6ywHHCKoro MO>KHO ea-
3B8Tb napaJI.OKCOM JI .JIHHHOH BOJIHbl. 06oHCHeHHe 38KJIJOl!aeTCH B
TOM, l!TO nocrpoeHHH CrpoiiKa oruocHTCH K cJiyllaJO, KOr.lla h/R.
cpaBHHMO C h/')..., B TO speMH K8K BblBOJlbl P3JieH npHMeHHMbl TOJlbKO
K cJiyl!aJO h(>. ~ h/R ~ I.
§ 16. napa,noKc «l>eppH
fopaano 6oJiee ee.naetteu napa.lloKc, KOTOpblM Mbl o6Haaehl
cl>eppe 3)' OTHOCHTCH K ceepX3BYKOBOMy o6reKaHHIO c «npncoe.1ui-
HeHHOH» y.llapHOH BOJIHOH H8KJIOHeHHOro Kpyroeoro Koeyca, OCb
KOToporo o6paayer yroJI. «pblcKaHbH» a c uanpaeJieHHeM retJeHHH,
KaK 6y.lleT DOK838HO B § 88, H3 rnnoTe3bl (C), § I, CJie.llyeT, l!TO
TaKoe TelleHHe .llOJl>KHO o6JI8.ll8Tb KOHU'leCKOtl CUMMerpuea. H03TO -
1) CM. Mason M., Gravity waves, 315 -320, Nat. Bu. Standards Circular
521, 1952, HnH rn. III H3 pa6011,1 Cornish'a, llHTHpOBaKHOil a npHMeqaHHH I) Ha
CTp. 41 .
2) St r u i k D. J ., Rendic. Lincel, I (1925), 522-527 . 06cy>1<.11eHHe napa-
. 1 1 .0Kca .llnHRHOil uonHbl CM. B [7]; Urse I I F. , Proc. Camb. Phil. Soc., 49 (1953),
685-694; BenjamIn T. B., LighthIII M. J.,
Proc. Roy. Soc., A224
(1954), 448-460.
3) Ferri A., NACA, Rep., 1045 (1961); CM. raK>Ke Ho It M., QIMAM,
7 (1054), 438-445.

§ 16. OapacJoKc <l>eppu
43
MY Mbl 6y..a .eM paccMaTpHBaTb u = U(<p, a) e cq>epnqeeKHX KOOP.UH ·
HaTax.
EeJIH OTO>K,D.eCTBHTb eooTBeTCTBYIOlUHe JIHHHH TOKa npH ueH-
rpaJibHOM npoeKTHpoeaHHH H3 eepWHHbl KOHyea, TO OHH eoeTaBSIT
O.D .HOnapaMeTpHqeCKOe ceMeHeTBO, KOTopoe exeMaTH4eeKH H306pa-
>KeHO ua pne. 5 . 3a HCKJJI04eHHeM JIHHHH TOKa, Jie>KalUHX s nJIOeKO·
eTH eHMMeTpHH, .l lJISI KOTOpblX O= 0, 'It, see JIHHHH TOKa CTpeMSJTeSJ
P II c. 5. napa.n;oKc <l>epp11.
K npe.11.eJibHOMY iHanpaeJieHHIO (a. -
a. 1t), T. e. see OHH CTpe-
MHTCH BJJHTbCSI B npHMYIO JIHHHIO TOKa, H.D .YlUYIO no KOHyey H eoeTa-
BJIHJOW.YIO eaHMeHbWHH yroJI a. -
a e .uanpasneuneM reqeHHSI. Ho,
e CHJIY ypaeeeeaii P3HKHHa - f10rouno, JJHHHHM roKa, nepeeeKaJO-
W.HM «npHCOe.D .HHeHHYIO> y.n.apHylO BOJJHY no.n . pa3JJH4HblMH yrna:.rn ,
eOOTBeTCTBYIOT pa3JIHllHble 3HalleHHff 3HTpOnHH. Do3TOMY U (<p, 6)
HMeeT oco6y,o TO~Ky B (a.- a, 1t), 4TO ettosa HapywaeT rnnoTe3y
(E) H3 § I. 3Ta oco6ettuocTb ..a .enaeT ttenpaeoMepHblM pa3JJO)Kettne
U (cp, B) no CTeneHSIM yrna pb1CK8HbSI a H B psi.n. <l>ypbe OTHOCHTeJlb·
Ho B. CJie..a .oeaTeJJbHO, Bb14HcnettHSI Konana .n.nsi oue11KH 3q>q>eKTOB
pb1CK8HbH, KOTOpble OCHOB8Hbl Ha TeopHH 803MyllleHHH, nenOJJb·
syiow.eft T8KHe pa3JJO)KeHHSI 1), He SIBJISIJOTeH CTpOrHMH. A ene.n.oea-
teJibHO, CTporo He o6oCHoaatt H napa.QoKc Konana (§ 11).
1) Stone A. H .. J . Math. Phys. MIT, 27 (1948); 67-81. [B11e roHKoro
s11xpesoro cno11 &6.riH3H Ko11yca pa3J10lKe1111e CToy11a npan11.11 ,110 . B [I 1•1 0110
aHallHTH'le<:KH npO.D .OJllKeHO BHYTPb IIHxpeeoro CJIOH, r .n;e OHO nepeXO.D .IIT B pa.3-
JIO>KeHl1~ BHJJJlera [12•1. -
llpuM. peo.)

44
I'It. I. Jlapai}ox:cbl H8BR3KOM T8'l8HUR
§ 17. Ilapa,l1,0KC TPOHHOH Y.l1.RPHOH BOJIHbI
B § 14 ynoMI-IHaJIOCI:> 0 TOM, '!TO ypaBHeHHH P3HKHHa - f10rc-
HHO BbIBO,UHTCH H3 3aKOHOB coxpaHeHHH. 3TH ypaBHemrn IlOKa3h, -
BaIOT, qTo B cJiyqae COBeprneHHOfO rasa OTHOIIIeHHH naBJiem1ii:, IlJIOT·
HOCTeii: u TeMnepaTyp pip', pip', TIT' no pa3Hhie cTopOHhI OT cra-
uuonapnoii: ynapnoii: BOJ!HhI 3aBHC5IT TOJlhKO OT O,UHOro napaMeTpa
(HHTeHCHBHOCTH CKa'!Ka HJIH 1!HCJia Maxa - CM. [15J, rJI. IV, § 4).
K:poMe HeKOTOpbIX HCKJIIOqeHHH, OTMel.JeHHhIX B KOH[(e § 14, 3Tl1
BhIBO,ll .hI no.nTBep}l{,ll.alOTCH 3KCnepHMeHTaJibHO, npuqeM Ha npaKTHKe
MO}l{HQ Ha6JIIO,UaTb y,napHhie BOJIHhI pa3JIH4HOH CHJ!hl.
He TaK o6CTOHT ,UeJio c «KpaTHhIMH» yJI.apHbIMH BOJIHaMH. Bcny-
IIae JJ:BOHHOH, «peryJIHpHO» OTpa}l{eHHOH y.napHOiJ BOJ!HhI ( CM.
puc. 6, a) u TpoiiHOH yJI.apHOH BOJ!HhI, HJIH Y-BOJIHhI Maxa (CM.
Ompa:nceHHctff
llaiJct!OUf051
Ompa:m:eHHa»
naaatoll{ctfl
CtfOllb:HCef!UJI
Ompa:m:eHHctfi
Ompa:JtCeHH/151
llaiJmou;aSI
a
6
P 11 c. 6 . OrpameH11e ·y.l(apm,1x BOJIH.
a - pery.irnpnoe 0Tpa>Ke1rne; 6 - Y -06pa3noe OTpa>Kemre no .Maxy.
p11c. 6, 6)' COOTBeTCTBYIOUJJie MareMaTHIIeCKHe pac4eThI B03MOiKHbl~
eCJIH npeJI.IIOJIO}l{HTh, '!TO B Ka2KJJ:OM H3 noJiyqaIOlll.HXC5I npH 3TOM
ceKTOpOB ( 1, 2, 3, 4) q>H3HIIeCKHe nepeMeHHbie npuHHMaIOT onpe·
neJieHHhie npe,neJihHhie 3HaIIeHHH B6JIH3H oco6on rotiKH B «Bepnrn-
He» yJI.apHOH BOJ!HhI. MHOrHe H3 TaKHX pactieTOB TaK2Ke IlO,ll .TBep-
>KJ(8IOTC5I 3KCnepuMeHTaJibHO, TaK IITO 3Ta Teopm1 SIB.;15leTC5I BecbMa
npas.n:ono .n:o6Hoii.
OmrnKo B cJiyqae «cJia6bIX» yJI.apHhIX BOJIH peryJIHpHbie orpa-
}l{eHust rrpoHCXOJJ:5IT rrpH yrnax naJI.eHHH HeCKOJlhKO 66JihllIHX, l .feM
sro .l{orrycKaeTCH reopueii, a no1tytte1tnbte npu pactte1e npeoe/lbHbte
8ftatteHUfl 0/lfl rpoiiftbtX yiJapHbtX 80/lH 3Hatture,1 .bft0 OT/lUl/,QIOTCJl OT

§ 18. 3Ha<1eHue ypaBHe1-1ua 3iiAepa
45
Ha6.1uoaaeMblX. 3To rrpoTHBOpetrne, KOTOpoe MO)KHO H33B3Tb rrapa-
.UOKCOM TpoiiHoii y.uapHoii BOJIHhl, 6b1JI0, rro-BH.UliMoMy, OTKpblTO
. [( )!(. <pOH Heii:MaHOM ( 1945 r.) .
0
He pa3 IIbITaJIHCb pa3pernHTb 3TOT rrapa.uoKC, KOTOpbIH, B03M0)1(-
HO, 5lBJI5leTC5l «rrapa.UOKCOM oco6oii TOqKH», T . e . IIOJiyqaeTC5l H3-3a
qpe3Mep1rn yrrpomeHHOH KapTHHbl JIOKaJibHOfO IIOBe,UeHl15l B6JIH3H
oco6oii TOlJKH. Ho ,!1.0 CHX rrop ne ,UaHO HH O)lHOfO y.UOBJieTBOpHTeJlh-
HOfO HCTOJIKOBaHirn l).
§ 18. 3HaqeHHe ypanHeHHH 3iiJiepa
Dpe.UbI.llYIUHe rrapa.UOKCbl IIOKa3b!B3IOT, qTO o6JiaCTb npHMeHH-
MOCTH ypaBHeHHH .3iiJJepa HMeeT neKOTOpbie orpaHtttieHH5I; O.UHaKO
3TH yp aBHeHH51 Bee eme 5IBJl5l!OTC5l OCHOBHbIM opy.UHeM rrpaKTHqe-
CKOH m.upoMeXaHHKH. TaK, OHH ,lla!OT B03MO)KHOCTb rrpH6JJH)KeHHO
BbJqHCJIHTb: 1) pacrrpe,lleJJenHe ,llaBJie1-rnii na JI060BOH IIOBepXHOCTH
nperr5ITCTBHl'i; 2) IIO,llbeMHYIO CHJIY KpbIJia caMOJieTa; 3) CHJ!bl rrpH
JI.BHllieHHH c «KaBHTaU.Hefi» (rn. III) H naJIHlJHH cTpyfi; 4) rH.ll -
poJI.HHaMHqeCKoe npoTHBOJI.eiicTBHe ycKopeHHIO rnepJI.oro TeJia B
lliH.llKOCTH ( «rrpttcoeJI.HHenHaH Macca», CM. rn. VI); 5) p acrrpoCTpa He-
~me
rpaBHT3U.HOHHblX BOJIH, BKJI10qa51 ceiiurn, npHJIHBbl H OTJIHBbI;
6) pacnpocTpaneHHe 3ByKa (a1<ycTHKa); 7) pacnpeJJ,eJieHHe JJ,aBJie-
HHH H CKOpOCTH TelJeHH5l B CBepX3BYKOBbIX COIIJiax H 8) CBepX3BY-
KOBOe Jio6oBoe corrponrnJie~rne .
DpH II0,!1 .06Hb!X pacqeTax neo6XO.llHMO HMeTb B EH.UY OIIHCaHHbie
Bbillle rrapaJI.OKCbI, a TaKllie 6oJib!llOe pa3Hoo6pa3He reqeHHl'i, y .llo-
BJJeTBOpHIOIUHX TeopHH neB5l3KOfO 06TeKaJ-JH5! rrpH HaJIHqHH 3aBHX-
peHHOCTH. B TeopeTHqecKoii rHJI.PO.UHHaMHKe 3TO pa3Hoo6pa3He HHO-
r.lla KaK 6b! OCTaeTC5l B reHH H3-3a Toro, qTo CJIHWKOM MHOro BHH-
MaHH5l Y.Ll€Jl5IIOT TeopeMaM cyu.1,ecrnoBaHH5l H €,llHHCTBeHHOCTH
KCTaTH CKa3aTb, rrpH ,llOKa3aTeJibCTBe TaKHX TeopeM qacTO HCXO.ll5IT
H3 nepeaJibHbIX .llOIIyIUeHHH. .3TO IIOJJ,qepKHBaeTC5l B 60JiblllHHCTBe
KHHr no «COBpeMeHHOii: fH.llPOJI.HHaMHKe» (HarrpHMep, B pa6oTax
[3] H [24]), rJI.e c caMoro HaqaJia y1<a3bIBaIOT, lJTO B03MOlliHa HeO.ll-
HosnaqnocTb pewenttii, a TaKllie OTMeqaJOT TaKHe Y.llHBHTeJibHhie
3KCnepHM€HT3JlbHble 5IBJI€HH5l, KaK norpaHHqHbie CJIOH H Typ6yJieHT-
HOCTb.
0,!I.HaKo HaJIHqHe TaKHX o6CT05lT€JlbCTB BOBCe He ,!I.OJl)KHO CHH-
>K3Tb 3HaqeHHe qHCTO MaTeMaTHqecKOH TeopHH H€B513KHX )KH]J,KO-
1
) CM. BIeakneyW. and Taub A. H., Revs Mod. Phys., 21 (W49),
587-605; [6], cTp. 342; [15], crp. 144; P o I ache k H. and Seeger R Phys
Rev., 84 (1951), 922-929, a TaKlKe [16] rn. 5 rex )Ke aBTOpoB u npHB~~eHHa,;
~~M JIHTepaTypa. I3oJJee HOBbie pa6orn J ah n R G., J. Fluid Mech., 2 ( I957) ·
-48
H Sternberg J., Physics of Fluids, 2(1959), 179-206 .
'

46
f /1. I . flapaUOKCbt HeBJl3KOZO Te'teHUJI
crett. TeopHSI B COCTOSIHHH yKa3aTb cymecrneHHhie B03MO)Kl(0CTII,
KOTOpble «3JJ.paBbIM CMblCJIOM» 6hlJIH 6hI OTBeprHyThl KaK a6cypJJ.Hhie.
HanpHMep, reopHSI yKa3hrnaer, qro Moryr 6bITh KpbtAb.n c npe-
He6pe:Heu.M .o MQAblM A0608blM conpOTUBAeHUeM. Xorn 3TOT HJJ.eaJI
noKa eme He JJ.OCTHrHYT, OH HBHJICH CTHMYJIOM Jl.JIH MHOrHX Ba>KHhIX
pa6or (cM. § 29).
HaKOHeu, H3 napa.noKca o6parHMOCTH cne.nyer B03MO)KHOCTh
roro, qro o6nacrh «MeprBor-0» B03JJ.yxa, HJIH «cJieJJ.» , MO>Ker 06pa-
3oearhcH enepeau UHJIHHJJ.pa. HaJIHqHe raKoii o6nacrn cnenano 6b1
B03MO>KHhlM o6reKaHHe KOHeqnoro UHJIHHJJ.pa raKHM )Ke IlOTOKOM,
KaK H H3BeCTHOe o6reKaHHe Teiinopa - MaKKOJia (§ 85) JJ.JIH KOHH -
qecKoro cnapHJJ.a . TaKoe reqenHe xapaKrepH3yercH reM, qro na 60-
Koeoii noeepXHOCTH KOHyca BCIOJJ.y IlOCTOHHHOe JJ.aBJieHHe. CorJiacno
reopHH «cJieJJ.OB» (rJI. III), reepJJ.hlH Konye MO>KHo 6h1JIO 6b1, ne
napylllaH paBHOBeCHH, 3aMeHHTb HJJ.eaJibHhlM HeBH3KHM 803.ll.YXOM
npH IlOCTOHHHOM H36hJTOqno!l.l JJ.aBJieHHH. MareMarHqecKH 3TO 03Ha-
qaer, llTO B HJJ.eaJibHOH >KHJJ.KOCTH B03MO>KHO o6reKaHHe IlJIOCKOrO
JJ.HCKa CBepX3BYKOBblM IlOTOKOM, npH KOTOpOM HeBHJJ.HMhIH KOHHqe-
CKHH B03JJ.Yll1HhlH 6apbep 38lllHlll.aer JJ.HCK or JJ.aBJieHHH B03JJ.yxa,
HaMHOro yMeHhllJaH JI060BOe conporHBJieHHe.
3JJ.paBblH CMblCJI H HHTYHUHH HeMeJJ.JieHHO OTBepraJOr B03MO>K-
HOCTb noJJ.o6noro retieHHH KaK JJ.O nenenocrH neycroHtIHBoro. Jiorn-
qecKaH OCHOBa 3JJ.eCh raKaH >Ke, KaK H B TOM cJiyqae, KOrJJ.a OTBep-
raIOr B03MO>KHOCTb cym.ecTBOBaHHH «CJieJJ.a» B o6nacrn BBepx no
retieHHJO. Ka>KercH B BhICllleii crenenH npaBJJ.onoJJ.o6HhlM, qro nanH-
tIHe npenHrCTBHH Jtaer ce6H 3Harh JIHlllh B o6nacrH, pacnono>Kennoif
BHH3 no relJeHHIO 1).
l1 ece >Ke B .ll.aHHOM cnyqae, no -BHJJ .HMOMY , CKopee ourn6aercsi
3JJ.paewi% CMblCJI, tieM MareMarHlJeCKHH BhlBOJJ.! CttHMKH, cJJ.eJiaHnhle
B A6epJJ.HHCKoii nayqno-Hc~JieJJ.oBareJihCKOH 6aJIJIHCTHtiecKoi% na6o-
paropHH ( CM. ¢poHTHCnHC), noKa3blB8IOT, llTO OlleHb TO HK a SI Hrna,
noMem.enHaH nepeJJ. JJ.HCKOM, JJ .eHCTBHTeJibHO cnoco6Ha C03JJ.aTb ne-
lJTO epoJJ.e nono6noro «a6cypJJ.Horo» retieHHH.
BBHJJ.Y 3TOro H JJ.pyrnx npHMepoB Mbl ClJHTaeM, '!TO y MareMa·
THKOB Her OCHOBaHHH JJ.yMaTh, 6y.nro B rHJJ.POJJ.HHaMHKe CHH>KaercSI
3HalJeHHe JJ.eJJ.yKUHH (npH conocraBJieHHH ee C «¢H3HlJeCKHMH BhlBO·
JJ.aMH»), eCAU YllHTblBaIOTCH re rHJJ.POJJ.HHaMHlJeCKHe napaJJ.OKChl,
KOTOpble BhlHB'HJIHCh B 3KCnepHMeHre.
1) Te >Ke C2Mble HHTYHTHBHble )lOra)lKH 6blJIH HCflOJlb3082Hbl B § 10, lJT06bl
Bbl)leJIHTb 6onee npe)lno'ITHTeJtbHOe «npocroe o6reKaHHe:. TOHKoro Kpb1.11a.

f.1aBall
nAPA)].OKCbl B513KOfO TPIEHH51
§ 19. YpaeHeHHH Haebe - CTOKca
HecMorp5! Ha 3H3'111TeJJbHYIO 06,1acrb np11MeHe1rnH ypaBHeHHH
3ttJiepa - Jlarpa,rn<a, HX , B006IUe fOBOp5!, 60JibWe He CtJHTalOT
np11eMJieMOH ocHoBott ,D.Jrn TeoperntJecKott rnnpo.D.HHaMHKH. BMecro
3THX ypasHe1rni% 11cnoJ1b3YIOTC5! ypasHeHttH Hasbe - CroKca, BbIBOL{
KOTOpblX Mb! ceil'lac KparKO H3JIO)KHM.
Bnepq:,1e 3TH ypastteHH51 6b!Jrn Bbme.neHbl Haebe ( 1822 r.) H
nyacCOHOM ( 1829 r.) . npHMeHHBWHMt:f ynpowem1y10 MOJJeKYJJHPHYIO
MO,!l.eJib ,!l.Jl5! ra30B, 'ITO npHBeJJO K BBe.D .eHHIO nOJIO)!{HTeJlbHOH 8513-
KOCTll μ > 0, KOTOpaH, KaK npe.nnoJJaraJJOCb, onHCbJBaeT MOJleKyJJHp-
HYIO .nmpcpy3HIO KOJIHtJeCTBa ,!J.Bll)KeHIIH.
0.nHaKO B HaCTOHIUee BpeMH o6wenptt3H8HO, 'ITO npOCTbie 3a-
KOHbl .ll.JIH Me)KMOJleKyJJHpHblX Cllll, npHIIHTble o6ottMII ytJeHbIMH,
6e3Ha.!J .e)KHO 11e coorneTcrny10r .nei%crn11TeJ1bHOCTH, oco6ettHo B
CJiytJae peaJJbHbIX )1{11,!l.KOCTeH. no3TOMY npHHUHnttaJibHO 6oJJee
npe.nnotJrnreJJbHbIM B 11acroH1uee epeM5! C'IHTaerC5l KOHnrnyaJibHbI1i
no.nxo.n CeH-BeHaHa (1843 r.) tt CroKca (1845 r.), KoropbIH no-
3BOJIHeT tt36e)KaTb yKa3aHHbIX npe.nnoJIO)KeHHH 1). Mb! HatJHeM c
H3JIO)KeHH51 noro KOHTHHY3JlbHOro (HJIH «MaKpOCKOTIHlfeCKOro»)
no.nxo ,ll.a .
TaKott no,ll .XOJJ . ocHoBaH 11a cpyH.rraMeHraJJbHOH rnnore3e, 3aKJI10-
tJa10IUettc51 B TOM, 'ITO K HanpH)KeHH51M ,ll.3BJiemrn, KOTOpbie pacota-
TPHBaJI 3iiJJep, Hy)KHO ,ll .o6aBIITb 8Jl.3KUe Hanp51)KeHHH, AUHeiiHO 33-
BHCHU.tHe OT c,wpocreu ae¢opAtal{UU. Htt)Ke npttBO.!lHTCH KJ)3TKOe
pe3JOMe npttMeHHeMbIX nptt 3TO~ apryMeHTOB.
u 3a)l.atJa 33KJIJOtJaeTCH B TOM, tJT06bI H3HTH CB513b Me)K,11.y Marpll-
uett (B513Kttx) HanpH)KeHHH P = llp;jll 11 MaTpttueil CKopocreil ,11.e-
cpopMau.1111 II au;;axj II. Kowtt nOK333Jl, 'ITO
.LlJl5! JII06011 cpe,lbl ,
ecJitt CtJHT 3Tb He,ll .onycTHMbIMH 6eCKOHetJHO 60Jlbllllle yrnoBbie
1
) noJIHYIO 15tt6JTHOrpa<jJHIO C~I. B (7], CTJ). 723; HCc. , eJlOBal!IIR CroKca 01.
: [13], T. I, crp. 7? H J13.1hWe; crp. 182 11 ,1a.1bwe. Xopomee coepcMPflHOC' 1n.10-
(IJ!l1"e CllM. Serr I n J. , I . Math. Mech., 8 ( 1959). 4 .59 -470 . (C~1. TaK)l{e [I'],
·
·--
pu4C. nepeB.)

48
[A. //. napaaoKCbt BJl3K0i!O 1e•1eHu.q
ycKopeHHH (rnnoTe3a (E) H3 § l), MaTpnua HanpmKemrn .ll .OJDKHa
6b!Tb CHMMeTpHl!HOH: Pii = Pii· C
. ll pyroii CTOpOHhl, MaTpHua CKO-
pocTH .ueqiopMaUHH eCTh cyMM8 KOCOCHMMeTpHtlHOH COCTaBJIHIOIUeii
II aui{ax,- auJ1ax1 II
2
COOTBeTCTBY!Olll.eH apamemno a6COJIIOTHO TBep.L1.oro TeJia, tt CHM-
MeTpHl!HOH COCT8BJIHIOIUeH
II aui1ax1 + au1/a.x, II
S=
2
.
TaK KaK nptt spamemrn rnep.L1 .oro Tena cpH3Hl!ecKaH .L1 .etpopMa-
UHH OTCYTCTByeT, TO CHMMeTpHl!H8H MaTpnua S Bbipa}KaeT HCTHH-
HYIO CKOpOCTb .netpopMaumt:.
CTpH HaJIH'IHH H30TpOITHOCTH fJl8BHbie OCH M8TpHU p H S .11:0Jl-
)l{Hbl COBIT8.ll .8Tb (Mbl H8ITOMHHaeM H3BeCTHYIO TeopeMy anre6pbl,
COrJiaCHO KOTOpoii BCHKYIO CHMMeTpHllHYIO MaTpttuy MO}KHO CBeCTif
K .ll.HaroHaJibHOMY BH.ll.Y nyTeM BpameHHH, npHBe.ll .H ee K COOTBeT-
CTBYIOIUHM «rJI8BHblM» OCHM KOopmrnaT). l{anee, H3 H8UJHX npe]l-
ITOJIOJKeHHil O JIHHeHHOCTH H H30TponHOCTH HeTpy.ll.HO ITOJIYllHTb, l!TO
OTHOCHTeJibHO rJiaBHbIX oceii cnpaBe.ll ,JIHBO COOTHOUieHHe
,d"
2μ au,
Pu=P-11. 1v u--a- -
x,
(1)
npH H8.D:Jle)KalUHX ITOCTOHHHhlX Ao H μ. I<poMe Toro, C y'leTOM CHM-
MeTpHH OTHOCHTeJibHO rJI8BHOH JI.H8fOHaJIH, MO}KHO 38ITHC8Tb pa-
BeHCTBO
au,
au1
p11= a.xi+ax, =0, ec;m i+j.
Bo3spamaHcb K o6meii cncTeMe KoopnttHaT, Mb!
.ny10w.tte OCHOBHbie ypaBHeHHH:
( au1 au1)
p,1= p'811 -
"J..divu- p. a.xi+ax, ,
KOTOphie 6yJI.eM npHMeHHTb HHJKe.
nony1.1ttM cne-
(1 *)
EcJIH He BBOJI.HTh npe.nnoJIO}KeHHH O HeC)KlfM8eMOCTH, TO Ol!eHb
TPYJI.HO nony1.1ttTh JI.JIH CHCTeMbI ypasHeHHH Haabe - CToKca Kop-
peKTHO IlOCT8BJieHHYIO Kpaeay10 38.ll .81 .!Y, YCJIOBHH KOTOpoii 6bIJIH 6bI
cl>H3Hl!ecKH COCTOHTeJihHblMH. npe>1<JI.e Bcero IJ8CTO 6hrnaeT HeH3-
BeCTH8 BeJIH1:iHH8 A (CM. § 33). CTOKC npo60B8JI npe.n:IIOJIO)l{HTb, l!TO
«BTOpaH BH3KOCTb» μ' o6pamaeTCH B HYJib:
p.' = "J..+!p.= O.
(2)

§ 20. PeaAbHbte 2a3bl u :RCuaKOCTU
49
9TO MO}KeT 6bITb BhIBe)leHO H3 KHHeTH'IeCKOH reop1m ,ll_Jl5l O)lHO-
aTOMHblX ra30B I). MO}KHO H ornn6ol!Hb!M crroco6oM BbIBeCTH ycJio-
BHe (2), onpeoeAua ,n_aaJiemrn p B BHlle (P11 + P22 + Paa) /3. Jlo-
BYillKa COCTOHT B TOM, l!TO He H3BeCTHO, orrpe,n _eJrneT JIH 3TO «,n _a -
BJieHHe» IIJIOTHOCTh cornaCHO repMO)lMHaMHI.JeCKOMY ypaBHeHHIO co-
CTO.HHH .51 p = p(p, T)' B KOTOpOM p H p 6epyTC5l H3 CTaTHI.JeCKHX
H3MepeHHH. ECJIH 3TO TaK, TO ycJIOBHe (2) HMeer MeCTO ([7],
crp. 718); B IIPOTHBHOM CJiytiae, Mb! He 3HaeM, KaK CBH3aTb repMo-
.lJ;HH8MHI.JeCKOe .!J,aBJieHne c TeH30poM HanpH}KeHHH II Pii II.
§ 20. PeaJihHhie ra3hl H :lKHJJ.KOCTH
CToMHMO CKa3aHHOfO, B peaJibHb!X }KH.ll.KOCT51X BeJIHI-IHHbl A H μ
H3MeHHIOTCH BMeCTe C TeMneparypoH T H naBJieHHeM p; Harrpn-
Mep, H3MeHeHH51 TeMne.paryphl HMe!OT 60JihllIOe 3HaI.JeHHe .!I.JIH
cMa3KH. B JIYI.JllleM cJiyI.Jae Mo:>1rno HaneHTbCH, I.JTO
'),.,, (p, T) H
μ(p, T) - o,UH03HaI.JHhie cpyHKUHH. LlJI5! roro I.JT06bI 3TY 3aBHCH-
MOCTh yI.JeCTb M8TeMaTHI.JeCKH, ypaBHeHHH HaBhe - CTOKCa Heo6xo-
.lI.HMO
.IJ .OIIOJIHHTb no MeHbllleH Mepe ypaBHeHHeM rerIJionpoBO.II.HOCTH .
3To .n .eJiaeT KpaeByIO 38.II.aI.Jy COBCeM He IIO!()laIOmeHrn pellleHHIO;
HO )la:>K:e H .nonOJIHeHHa5l CHCTeMa qrn3HI.JeCKH He TOI.JHa, TaK KaK MbI
IIpeHe6pernH H3JiyI.JemreM. (TaKoe rrpeHe6pe}Kett11e secbMa IIpaB.n:o-
no.n:o6Ho B cHJIY rmrore3bI (B) H3 § 1.)
0JJ.HaKo ABH}KeH11e peaJibHbIX }KH)lKOcTeH cBH3aHo TI c APYrHMH
<pH3HI.JeCKHMH 3cpcpeKTaMH, KOTOpb!e He yI.JHTbIB8JIHCh HH HaBbe, HH
CroKCOM. TaK, B peaJihHbIX ra3aX npn rnnep3BYKOBhIX cKopocrnx
TeI.JeHH.51 B8}KHYIO pOJib HrpaIOT 3cpcpeKT peJiaKC3UHH, MOJieKyJIHpHaH
J{Hccoun:aun5! 11 nottn3aUHH 2). 6yAymn11 cneunaJIHCT no rnJJ.poMe-
xaHn:Ke, KOTOpoMy npH)leTCH HMeTb )leJIO C 3a)laI.JaMH, CB .H3aHHblMH
co CIIYTHHKaMH H HX B03Bpame1rneM, )lOJI:>K:eH )lOllOJIHHTeJibHO K
ypaBHeHH.HM HaBbe - CTOKCa xopOllIO 03HaKOMHTbC5I C XHMHI.Je-
CKOH KHHeTHKOH.
CToJJ.06HbIM o6pa3oM 6HI.JOM nepBbIX CBepx3BYKOBhIX a3pomrnaMH-
qecKn:x Tpy6 6b!JIH y)lapHbie BOJ!Hbl, B03HHKaBillHe H3·3a KOH)leH-
caun:n: BO)l51HbIX napos s B03)lyxe - eme O)lHa «cKpbITa.H IIepeMeH-
Ham>, KOTopy10 HrHOpnpyIOT npH IIOCT8HOBKe 38)lal! no HaBbe H
CTOKcy; CM. [16, rJI. 5].
3KcnepnMettraJibHO 6hIJio o6ttapy}Ketto, I.JTO 3aryxamre 3ByKa
B )Klf)lKocrnx II ra3ax - HBJieHHe, KOTopoe orrpeJJ.eJieHHO npe.n:cKa-
1) Max we II J. C., Phil. Trans., 157 (1867), 49-88; lI en Me H C., Ka Y·
JI II Hr T., MaTeMaTwrecKaH Teop11H Heo.rrnopo.u;HhIX ra3os, HJI, M., 1961. .ll)oreM
llOKaaan, 'ITO H3 BTOporo HatJaJia TepMO.llHH3MHKH CJie.u;yeT μ' > 0.
p 2) CM. Light hi 11 M... J ., s «Surveys in Mechanics», Cambridge Univ.
ress, 1956; 250-351; Xe H3 Y. LI,.
11 TIpo6cr uH P. <I>., TeopttH rnnep-
8BYKOBbIX TetJeHHii, M., HJI, 1962.

50
f" . l. If. Dapa<iOKChl 8!/:JKO?O re•tl'HU!I
-----·-------------~
3bIBaeTcfl J11060A ro,1110A TeopHeH c>KHMaeMbIX B513KHX >KHJJ.KOCTeH,-
B 3HatJHTeJibHOH Mepe 3aBHCHT OT MOJ1e1,y .151pHblX 3QHpeKTOB (pe-
JiaKcauHH), CM . § 33.
BBHJJ.Y ynoMHHYTblX BblWe TPYJJ.HOCTeH, no-BHJJ.HMOMY, pa3yMHO
orpaHHlJHTbCS! paCCMOTpeHHeM HeC:JICUMaeMbtX Bfl3KHX >KHJJ.KOCTeH,
HO H B 3TOM cJiytJae H3BeCTHbl pa3JIHlJHble aHOMaJIHH. TaK, r.rnorne
>KHJJ.KOCTH , COCTOfllUHe H3 JJ.JIHHHblX MOJieKyJrnpHblX uenotJeK HJIH
co.11 .ep)l{aW.He fJIHHHCTble B3BeCH, Ha3bIBalOTCfl «HeHblOTOHOBblMH» -
TaKOB ynoTpe6J1f1€MblH B HaCTOHW.ee BpeMfl pa6ot!Hif TepMHH J].Jlfl
)l(HJJ.KOCTeH, Koropb1e He y.11 .onJJernopfllOT ypanHeHHf!M Hanbe - CTo-
Kca. CnepxTeKytJeCTb >KHJJ.KOro remrn - ew.e OJJ.HO 51BJl€HHe, KOTO-
poe He cornacyeTcH c TeopHeH CroKca 1).
§ 21. HeC}l{HMaeMble BH3KHe }l{HJl,KOCTH
BBHJJ.Y TPY .ilHOCTeH, onHcaHHbIX B § 20, OCHOBHoe BHHMaHHe Ma-
TeMaTHKOB 6blJIO cocpeJJ.OTOtJeHO Ha ypaBHeHHf!X Hanbe - CTOKCa
JJ.JIH HeC>KHMaeMblX Bfl3KHX )KHJJ.KOCTeH B npe.11 .noJJO)KeHHH, lJTO BeJIH-
lJHHbl μ H p MO)l{HO ClJHTaTb npHMepHO noCTOf!HHblMH. 6oJibWHHCTBO
cneuHaJIHCTOB ClJHTaeT, lJTO TeopeTUtteCKQJl. i!UdpOdUHQMWW, OCHO-
BbIBalOW.af!Cfl Ha ypanHeHHHX Hanhe- CroKca, .1 1.aeT JJ.OBOJibHO TOtJ-
Hoe npH6JIH)f{eHHe JJ.HHaMHKH peaAbHbtX >KHJJ.KOCTeH, eCJIH lJHCJIO
Maxa M HaCTOJibKO Mano, lJTO MO>KHO npeHe6petJb 3cpcpeKTaMH C}l{H-
MaeMOCTH. OttH ysepeHbI B TOM, tJTO (nepecppa3HPYfl JiarpaH)l{a)
«ecnH 6bI ypaBHeHHfl Hanbe - CroKca 6b1J1H HHTerpHpyeMbJ, To npH
MaJiblX lJHCJiax Maxa MO>KHO 6b!JIO 6bl noJIHOCTbJO onpe.11 .eJIHTb see
JJ.BH>KeHHfl )KHJJ.KOCTH» ( cp. § l). Jlm1 Toro lJT06bI HCCJieJ].OBaTb, Ha-
CKOJlbKO o6oCHOB,rna TaKaf! ysepeHHOCTb, Mb! npeo6pa3yeM CHatJaJia
3TH ypaBHeHHfl K 6onee y.11.06HOMY BHJJ.y .
06'beJJ.HHHB ypaBHeHHfl ( l *) c ycJIOBHeM H€C)KH MaeMOCTH ( cpop-
MyJia ( l ') H3 rJI. J), Mb! OOJIYtJH M ypaBH€HHe
(
iJu1
OUJ)
P11= Pa11- 11- ax1+ax, ·
ECJIH npH 06bJtJHOM Bb!BOJ].e ypaBH€HHH JJ.BH .lK€HHfl ytJeCT.b
ttJieHbl, co.11.ep.lKaW.He BeJIHlJHHY μ, TO nonytJHM BM€CTO ypaBH€HH51
9iinepa (2) B rn. I cne.11 .yJ01Uee ypaBHeHHe:
~1
μ
Dt+Pgradp=g+vV
2
u, v=P.
(3)
BMecTe c ypasHeHHeM
div U=O
(4)
1) CM. London F.,
Superfluids, Wiley, 1954. CTp. 6 -13; Donne 1-
1y R J., Phys. RetJ.,
109 (1958), 1461-1461 11 np11Bell.eHHY10 raM JJHreparypy.

§ 21. HecJfCUAlaeAlbte BJl3KUe 3/CUOKOCTU
51
ypaeueutte (3) onpe.ueJIS-rer ( HblOTOHoey) HecJ1CuMaeMy10 BR3KYHJ
o,cud,cocrb.
Jlerno y•1eCTb, KaK .neiicrnyer CHJJa TSDKeCTH Ha TBep.uoe TeJJO,
norpy)KeHHOe B no.uo6Hyl0 »<H]lKOCTb, HCil0Jlb3YH CJJe]lylO~HH npHH·
UHn 1).
Teo p e Ma I. J/..1H BH3Koii J/Cua,cocrn c nocrORHHOU nAOTHOCTblO po
2paBUTal{UOHHblU 3¢(/JeKT 3K8UBOAeHTeH HQAOJ/CeHUIO zuapocra·
TU'leCKOZO aaeAeHUJI poG.
)loKa3areJJhcTao. noJJaraH a
= - aa;axi, r.ue G - rpae,nauHOHHblH
e pe3yJJbTare ypaeHeHHe
ypaeHeHHH (3) gi =
noreHUHaJJ, Mbl noJJyqHM
Du1
l
~
2
Dt +Pogradp= ~vu1,
(5)
n p e .u o crepe»< e H He. 3aMeTHM, qrn npeo6pa3oBaHHe reo-
peMbl I He coxpam1er 06b1quoro Kpaeeoro yc.' !OBH H
Ha «cao6oJJ.HOH
noeepxHOCTH» p = const JlJIH rpaHHUbI ra3-»<H.llKOCTh. CJienoaa·
TeJJbHO, OHO 6ecnoJie3HO npH H3y4eHHH BOJIH Ha noeepXHOCTH H Ka-
BHTaUHH (rn. III).
)];mi: roro qro6bl noJJytIHTh H3 ypaBHeHHH (3) H (4) KoppeKTHO
nocTaBJJeHHYIO Kpaeey10 3a.lla4y, BMeCTO ypaBHeHHH (7) rJJ. I BBe·
)leM Kpaeaoe ycJJOBHe npHJJHnaHHH 2 ) B CJJeJJ.yIOIUeM BH.IJ .e:
u (x) = 0 113 JII06oll Q)HKO!pOB3HHO!l rpaHHU.e .
(6)
(Ha .U .BH>KYIUHXCH rpa1muax U (x) = v(x) - CKOpOCTH JJ.BH»<eHHH
rpaHHUhl, e ro epeMH KaK a HeBH3KOM cJiy4ae rpe6yerrn, qro6bI
(51,1na HenpepbIBHOH TOJlbKO HOpMaJibH851 COCTaBJIHIOIU8H CKOpOCTH.)
HanoMHHM T8K»<e OCHOBHOH npHHUHn no;J.06HH .
TeopeMa 2. DycTb U=f(x, t) yaoaAerBOpReT ypaBHeHURAt
(3), (4) u (6) npu p = canst. EcAu aeAU'lUHbt V, L, v, V', L', v'
nocroRHHbte u ra,cue, 'ITO V L/v = V'l'/v', 10
(V') (lx LV't)
u'=g(x, t)= v f ---y,, -y,v
(7)
TOIC3/Ce yaoaAeTBOpHeT ypaaHeHUflM (3), (4) u
saMeHa p Ha p' up Hap', npu'leM
(6), zae npoeeaeHa
p'V '2
p'-
p'O = p°V2 (p- pO).
(8)
1) Ka r m a n Th., J. Aer. Sci., 8 ( I94 I), 337-356. B tteel!3KOM cJJytJae :.ror
pesynLrar noJJytJeH llaJiaM6epoM, Theorie de la resistance des fluides, craTbH 48,
56, 91, H AeaHUHHII (1807).
2) 0 3Ha•1eHHH 3TOro yc.10Bllll Mbl CK3)KeM B § 3,~.

52
r A. JI. IlapaiJoKcbt BJl3Kow retteHuR
JI.pyrnMH cJioBaMH, mo6oe II3MeHeHHe MacnITa6a (B rrpocTpaH-
crne II BpeMem1), coxpa HSIJOlliee HeH3MeHHbIM tf,UCAO PeiiHOAbaca
VL/v = pVLJμ = Re, nepeao,n,11T Hec:>KHMaeMbie TeIIemrn, y,n.oaJie·
TBOp.HIOIII .He ypaBHeHH.HM Haahe- CTOKCa, B perneHH5I Tex }Ke ca-
MbIX ypaBHeHHH.
BU wa E=-:E-=· --ES=l=f=f=f=l=l=l==Sf=E=l==l==SEE
40 60μ, ... ,__-+ --+ --+- -+--+--+ --+- -+--+--+--+ --+- -+--+--+ --+- -+--t--t-~
zo " f-....
8 JOl=t=~~=i=t=~=i=:!=l=t=t==t=!:=i==t=!=t~
4 61--t~+ --f- -~ '"'-'l---t- -+ -- -t --+- -+ --+- -+ --+- --t- -+ --+ ---t- -+ - -+ --t- ~
~~ l,1--1~+ --f- -f- -~
.......
~ ~r--=- -~-- +--+--+--+-+- -+---+--+--l--+--+--l-~
O,B o~l=±::±::±=±=±=l:=t==l=::1::=.1==±::±:-~~~==l=~~.::::t=:J,
t.f•
D,4
•
D,Z1--1~+--f- -f- -+ - -+- -+~+--f- -l~f--l - --+~+---1~+~l--l -- -+~~_,,,~
a,, .........~..._...._.__..__....._.~.__~2,,...__..__._J_.___._~
10
- 4- -_ .._ __ ._,s ,..._....... .~ 6
W
125/0
W
10
ID
m
800 WDD
400 600
80
40
8
4
200
/OD
60
20
ID
6
2
~: /a
w
{J,OB ao·J
0,04
'
0,02
~-·
l'c
'
Re .. ud/v
PH c. 7a. CD (Re) ,nJJS! I.\HJJHH,npa.
..
·t+
I
....
"-
I'. .
"-
........
I'- _
0
·%.oz 0,080,20,1013 z 1 B 20 80 200 800 4000 20000
He
PH c. 76 . CD (Re) ,nJJll cqiepbl.
I\
I~
400000 4 000 000
c JI e Ac TB II e. EcJlH AB a CTaUHOHapHhlX TetJeHH5l YJ.lOBJieTBo·
pmoT KpaeBOH aa,n.aqe (3), (4), (6) npH OJI.HOM H TOM iKe tJHCJie

§ 22. llapaiJOKC Hel!HG,IUTU 11HOCTU
53
------
PeiiHOJlb,,lCa II eCJIH :na KpaeBaH 33.ll.34a MaTeMaTH4eCK!I KOppeKT·
HO nocTal3.1eHa, TO o6TeKaeMbie 3TJIMII Te4eH1151MH Te.TJa .ll.OJl)l{Hbl
HMeTb O.ll .llH II TOT )Ke K0:1cpcpmu1e1n JI06osoro conpOTHBJICHHH
Cv(Re).
KJiacc11t1ernoe 3KcnepHMeHraJibHOe nonrnep)l{,UeHHe .ua ,rnoro
CJie,UCTBHH (Ho He o6wrnreJibHO rex rHfIOTC3, KOTOpbie npH 3TOM
6b!JIH HCTIOJ!b30B3HbI!) flOK333HO Ha pHC. 7a 11 76, r,lle np1rne.ueHbl
K03cpcjmuneHTLI .106osoro conpoTHBJieHH5l COOTBeTCTBCHHO .ll.Jl5l Ull·
mrn.upa 11 ccpep1,1. Ensa JIU MO)KHO 6w10 npennoJio>KHTb cymecrno-
Bam1e 3Tl!X 3c:lMe'!aTe.nbHblX KpllBb!X, eCJ!ll 6bl CBOl!CTBa B5l3KOCTJI He
6wm YK333Hbl B T04HOii MaTeM3TH4eCKOH cjlopMyJIHpOBKe!
npencras.TJemrn, Jiearnw.ne B ocHose reopeMbI 2, 6y.uyT nonpo6110
npoaH3.TJ!i311pOB3Hbl B § 71.
§ 22. napa)I.OKC neanaJIUTH4HOCTH
J1arpaH)I{ nocrpo11JI nepsoe ,noKa3areJibCTBo Toro, 4TO B HeBic!3·
KOH )l{l!,ll .KOCTH 33BHXpeHHOCTb 43CTHUhl )1{11.ll .KOCTH 51BJI5leTC51 nep·
MaHeJ-ITHOii. K CO)K3JICHHIO, ,UOK333TCJ!hCTBO J1arpaH)l{a, KaK ITOKa·
33.TJ CTOKC ([13], T. I, crp. 106 -112), orm1604HO. OHO 0.Il.HllaKOBO
npHMCHHMO II K o6JiaCT5lM B B513K01! )1{11,UKOCTI-I, r .ue 3Ta 33BHxpeH·
HOCTb HenepMGfieHTHal Ow116Ka 33KJ!l043JI3Cb B TOM, 4TO CKOpOCTb
H 33BHXpeHHOCTb npe.unoJiaraJIIICb aHG/lUTll'leCIWMU cpyHKI-lH5lMll
BpeMeHH.
Ecm1 3TO npHHHTb (cornacno nrnore3e (E) 113 § 1), ro MO>KHO
paccy)l{.rr .arb CJie,UyIOlUHM o6pa30M. OcHOBHOe ypaBHCHHe (3) 3KBH·
B3JICHTHO (eCJIH TIP'HMCHHTb onepau1110 rot K o6eHM ero 4aCT51M)
ypaBHCHl!IO
(9)
OTHOCHTeJibHO 33BttXpeHHOCTH S = V X U. C IlOMOlUblO HC33BHCH·
MbIX nepeMCHHb!X JlarpaH)l{a a H t, r).I ,e a OTHOCHTCH K ,UBH)l{yw.eiic51
43CTHUe, TaK 'ITO a/iJt (a cjJHKCHp0133IW) CCTb D/Dt, Mb! MO)l{CM
npeo6pa3oBaTb t1acrn1,1e npOH3BO.UHbie no cpopMyJiaM
u
~
a
axk =~Akj(a,t)aaj •
0pHMCH5l5:1 3TY onepauu10 K ypaBHeHHIO (9), noJiyqaeM COOTHO·
IllCH He
mi~
iJu;
Dt + ~~k(a, t)Akj(a. t) iJa· =vV2
~;+(S · V) ui.
(10)
J
flocJie,n:osaTeJibHo .D .!!cpcpepemLlf PYB coornorneHne ( 1O) no spe·
Menu f npn TIOCT05lHl!OM a, I!O .TIYlJHM nocJie,n:osareJihHOCTb HBHb!X
Bb1pa.iKeH11ii .ll:JI5l D'!~)Dtn. Jlerno noKa3aTh, 'ITO Ka)KJJ.bitt 4J1eH e

54
I'A. II. llapaiJOKCbl 6Jl3KOW Te'leHUJI
----
-- --- ---- ---- --- ---- ---- ---
l{a)KJ.lOM TaKOM Bh1pa)l{eH1rn COJJ.ep)l{HT B Ka4eCTBe MHO)KHTeJISI JIH60
~ i, J11160 V2
~ ; , JIH6o 0.ll.HY H3 npOH3BOJ.lHblX
DD
Dn-lc D n-1
~if t, ,,,,
~tf t
H T. J.l.
no3TOMY, npeJJ.nOJiaraSI see cpyHKUHH JJ.Hcpcpepem.tttpyeMhIMH 6ec-
KOHe,rnoe 4HCJio pa3, HH .ll.YKUHei'! no n
noJiy4aeM, 4TO ace
Dn'¢.dDtn=0 1).
HaJIH4He BSI3KOCTH npoSIBJISieTCSI B 'IJI_euax C npoCTpaHCTBeH-
HblMH npoH3BO)lHhlMH OT 38BHXpeHHOCTH . .UJJ51 HeBSI3KOH >KH.ll.KOCTH
H3lJaJibH851 3aBHXpeHHOCTh s(a, 0) = 0 B JII060H T04Ke X(a, 0)
o6ecnetJHBaeT TO, lJTO nee D"'MDtn (a, 0) = 0 B Tex }Ke TOtJKax, B TO
BpeM51 KaK B B513KOH )l{H)lKOCTH .ll .Jl51 o6pameHH51 B HYJih npocTpaH-
CTBeHHhlX npOH3BO)lHblX OT 3aBHXpeHHOCTH Tpe6yeTCSI OTCYTCTBHe
38BHXpeHHOCTH B HeKOTOpoi'I OKpecrnoCTH T04KH x(a, 0).
l1 B TOM H B ,npyrOM CJiy4ae, eCJIH cpyHKUHSI s(a, i) QHQAUTU'le-
CKUR no t, TO OHa TO>KJJ.eCTBeHHO o6pamaeTC51 B HYJib, TaK KaK nee
' IJieHhl ee pa3JIO>KeHH51 B p51,ll. TettJiopa (no t) TO)l{)leCTBeHHO paBHhl
Hymo. 2ho npHBOJ.lHT K CJie)lyJOll.leMy napaJI.OKCY 2 ).
n apa.!l.0KC HeaHaJIHTH4HOCTH. /J.ARTOZOllT06bt06-
;!aCTb JJCU8KOCTU, ,wxo8R.U{QRCR BHQllQAe 8 COCTORHUU noKOR ( UAll
8 6e3BUXpeBO.M 8BUJ/CeH.UU), CTQJlQ 3QBllXpeHHOU, OHa aoAJJCHa yJJCe
/l.MeTb 3QBUXpeHHOCTb, ,wropaR RB,lfl .eTCJI HeaH.a ,1UTUlleCKOU </JYHK-
l{lleU apeMeHu.
§ 23. Cy ~eCTBOBaHHe H eAHHCTBeHHOCTb
npe)l{)le 4eM BhlSICHHTb npHro)lHOCTb ypaBHeHHH HaBbe - CTOK-
ca !lJISI onHC3HH51 MexaHHKH peaJibHblX ( 11ec>KHM aeMblX) )KH)lKOCTeH,
HaM CJieJJ.OBaJJo 6bl y6eJ).HTbC51 B TOM, lJTO c HX nOMOUlblO MO)l{HO
<pOPMYJIHpOB3Tb <pH3HlJeCKH eCTecTBeHHble KpaeBble 33)lalJH, KOTO-
pble MaTeMaTHlJeCKH OK33blBalOTCSI KoppeKTHO noCTaBJieHHblMH ( CM.
TeopeMy 2, cJie.zi.crnHe). To ecTh Mb! !lOJI)KHbI HMeTb reopeMhI cyw.e -
CTB0BaH1rn H eJ).HHCTBeHHOCTH, KOTOpble .no CHX nop JlOKa3blB3JIHCb
TOJlbKO npH BeCbMa orpattttlJeHHblX .nonymeHHSIX.
4TO Kacaerrn 33Jla4H Kown (3a)la4H c Ha4aJihHblMH ycJIOBHH-
MH)' TO JlJl51 Hee cymecTBOBaHHe H e,nHHCTBeHHOCTb 6bIJIH )lOKa3aHbl
JI.JISI CJiy4aeB IlJIOCKHX H ocecHMMerpH4HblX Te'leHHH B npe)lnOJIO)l{e-
HHH KOHe'- IIIOCTH nOJIHOH 3HeprHH. npH J.lOK838TeJibCTBe HCnOJib·
1) EcJJH nepBOH8'18JlbHO ; = 0. -
llpUM. nepe6.
2) D.u hem P., Traite d'Energetique, T. 2, cTp. 121; Truesdell C., Ki·
nematics of vorticity, Indiana Univ. Press, 1954, § 104.

.§ 24. Te~cHue l7 yaJeci,z_q
55
3yeTCH ypaBHeHHe (9)' KOTOpoe ,' J.J151 TIJIOCJ<Oro Te'leHHH II Meer ynpo-
meHHblH BHA:
D(
" '''
Dt = '1v·1 .,,
(11)
0,D ,HaKO B npoCTpaHCTBeHHOM cJiy4ae ,D ,a}Ke AJIH KOHe4HOll nOJI·
HOH 3Heprntt 6b!JIO .UOKa3aHO TOJibKO cyw.ecTBOBaHHe - H TO JIHWb
. UJ I H orpa HH4eHHblX HHTepBaJIOB speMeHH 1). Xorn npe.unoJIO}KeHHe
O KOHe4HOCTH TIOJIHOH 3Heprntt, sepoHTHO, MO}KeT 6bITb OCJia6Jie-
HO, -
TIO}KaJiyii,
.UO C TaT04HblM MO}KeT OKa3aTbCH orpaHH4eHHOCTb
cKopocrn, -
E. Xoncp' 2 ) noKa3aJI, 'ITO 3a.ua4a Kowtt .u.rrn ypaaHeHHH
Hasbe - CrnKca He HBJJHeTCH KoppeKTHO noCTaBJJe1rnoii, ecm1 JJ.O ·
nycTHTb, '! T O C yseJIH4eHHeM pacCTOHHHH OT Ha4aJia KOOp.UHHaT CKO-
poCTb B03pacTaeT JIHHelfHO, a .uaBJieHHe - KBa.upaTH4HO.
B CTaUHOHapHOM cJiyqae TeopeMbl cyw.ecrnoBaHHH .UOKa3aHbl
. UJ I H o6TeKaHHH npenHTCTBHii npOH3BOJibHOH cpopMbl KaK B TIJIOCKO·
CTH, TaK H B npoCTpaHCTBe, HO H.e
.U OKa3aHbl TeopeMbl e.UHHCTBeH-
HOCTII. EcJJH paccMaTpHBaeMble CTaUHOHapHble Te4eHHH HBJIHIOTCH
e.UHHCTBeHHblMH, TO npH 6oJibWHX 4HCJJax PeiiHOJib.UCa OIIH cpH3H·
4eCKH Heycroii.'lUBbt; 3TO 51BHO CJie.uyeT H3 napa.uoKca Typ6yJieHT·
l:IOCT!I (§ 25) 3) .
§ 24. TelfeHHe nya3eitJU1
YpaBHeHHH HaBbe - CrnKca, KaK H ypaBHeHHH 311Jiepa Bo ape-
1v1e11a Jlarpatt}Ka, y .uaJIOCb TIOKa npOHHTerpttpoBaTb JIHWb B HeCKOJib·
KHX cJiy4aHX. no3TOMY corJiaCOBaHHe C 3KCnepHMeHT' .)M B 3THX He-
MHOrHX cJiy4aHX HMeeT npHHUHTIHaJibHOe 311a4e1me.
0.UHIJM H3 TaKHX cJiyqaeB HBJJHeTCH Te4eHHe }KH.UKOCTH B
.UJ IHH·
HOH npHMOii Tpy6e, nonepe4HOe ce4e11He KOTOpoii eCTb Kpyr
TIOCTOHHHOro pa.uttyca c. nycTb x o603Ha4aeT paccTOHHHe, H3Me-
pHeMoe B.UOJJb Tpy6bI, a r - paCCTOHHHe OT OCH Tpy6bl. B 3T!IX
UHJIHH.UplJLieCKHX KOOp.UHHaTax Ux, Ur H U.ri nyCTb o6o3Ha4a!OT co-
OTBeTCTBeHHO oceByIO, pa.uHaJJbHYIO H TpaHCBepcaJibHYIO COCTaBJIHIO·
ll~He cKopocrn.
Te Op e M a 3. EouHCTBeHHO 803MO:JICHblMU peut.eHURMU CU.CTeMbl
(3), (4), (6), 06Aaoa10U{UMU. npeo,wAaiaeMoii. cuMMeTpueu ( crn-
1) Leray J. , ! . de Math., 12(1933), 1-82; raM lKe 13(1934), 331-418;
Acta Math., 63 (1934), 193-248; Hopf E., Math. Nachr., 4 (1951), 213-231 .
CM. TaKiKe ,lloJIHJJ.3e )l. E ., llMM, 12 (1948), 165-180 H 19 (1955), 76·1.
2) Hop f E., !. Rat. Mech. Anal., I (1952), I07.
3) 3Ha'IIITeJibHble pe3yJibT3Tbl OTHOCHTeJibHO cyll.(eCTBOB3Hll!I H ycroii'IHBOCTH
peweuHii cra1.mouapHbtX H uecral\HOHapHbtX Kpaesb1x 3a, 'l,a 'I JI .JI!! ypasueHHii
Hasbe - CroKca no.~y4eHbl B p!!JJ.e pa6or 0. A. Jla.a;btiKeHcKoi\ 11 ee corpy.a;HH·
KOB. CM. JI a A bl)!( e H c Kall 0. A ., MaTeMaTH'!eCKHe aonpocbl AIIH3MHKH Bll3KOil
HeCiKHMaeMoii JKH,11,KOCTH, <l>H3M3TrH3, M.,
1961. -
llpuM. nepee.

56
L1. If. [/apaoOKCbt BR3KO.?O Te~eHU!I
4uo1-tap1-tbte retteHuH. BH.3KOt1 :JICUOKocru a Kpy2.zot1 rpy6e), J1.B1lJ/.10TCJI
ret.teH/IJ/. llya3eU/lJ/., onpeoe/lJleMbte qJOpMyllaMU
Ux=a(c2 -r2 ),
u,=U~=O.
(12)
JJ. o Ka 3 are JI b c Tao. CTpeJI.noJJo)KeHHe o crauttoHapHOCTH Te·
qeHH51 03HaqaeT, qTQ qiyHKUHSI u = u (x, r, 6) He 38BHCHT OT ape-
MeHH t. KpoMe Toro, ycJJOBHH 38JI.aqH HHBapttaHTHbl OTHOCHTeJJbHO
OTpaJKeHHSI B JJI06ol'i IlJJOCKOCTH, npoXOJI.HIUeH qepe3 OCb Tpy6b1;
reqeHHe HMeeT :ny CHMMeTpH!O Tar.II.a H TOJlhKO Tor.II.a, KOr.na
Ua = 0, Ux = f(x, r), Ur= g(x, r). CornacHO H.aWIIM ycJJOBHHM, . llOJJ·
JKHa 6b!Th T8K)Ke HHBapttaHTHOCTh OTHOCHTeJJhHO npOH3BOJJbHOrO
nepenoca BJI.OJJh ocH Tpy6hI. A TaK KaK v npeJI.noJJaraeTcSI He 3aBH·
C51IUHM OT Jl.8BJJemrn, TO 3TO )Ke OTHOCHTC51 H K ypaBHeHHHM (3) H
(4). CHMMeTpHH OTHOCHTeJJbHO nepettoca 3KBHBaJJeHTHa COOTHOWe-
HHHM Ux = f(r), Ur= g(r). l-fa 3THX COOTHOWeHHft H H3 yCJJOBH51
(6) cJJeJI.yeT, 'ITO div u = d[rg(r)]/dr = 0, OTKyJI.a g(r) = C/r = 0,
raKKaKHaOCHg(r)=0.
Teneph, noJJaraH g = 0, cornactto TeopeMe 1, Mb! HCTIOJJb3yeM
ypaBHeHHe (3). nocKOJlhKY U2 = U3 = 0, HMeeM p = p (x). PaccMa-
TpHBa51 CJJyqa)'i i = 1 (c OJI.HOH KOOpJI.HH8TOi-i x)' noJJyqaeM COOTHO·
weHHe
p' (x) = p.V 2ux = 11- I/" (r) + r1j(r)J.
TaK KaK JJeBa51 qaCTh He 38BHCHT OT r, TO npaaaH qaCTb T8KJKe
He Jl.0JJ)KH8 3aBHCeTb OT r. TaKHM o6pa30M, (rf')' = rf" + f' = kr
1
JI.JJH HeKoroporo TIOCTOHHHOro k, H rf' = 2 kr2 + K. 3ro .n .aer KOHeq-
HOe 3HaqeHHe Ux = f (r) npH r = 0, TOJJbKO ecnn K = O; cJJeJI.oaa-
1
TeJJhHO, f' = kr/2 H f (r) = 4 kr2 + b. ]J.JJ51 roro qr06bI yJI.OBJJeTBO·
p51JJOCb ycnoatte npttmmamrn (6) Ha rpaHHUe,
. 11 .0JJ}KHO 6bITb
u,.=a(c2-
,2), qro 3aaeprnaeT JI.OKa3areJJbCTBO reopeMbI.
noJI.CTaBJJHH nonyqeHHbie Bbipa)KeHHH a ypaBHeHHe (3), Mbl no·
JJyqaeM KJJaCCHqecKHH pe3yJJbTaT, qro rpa)l.HeHT Jl.8BJJeHH51 paaeH
dp
BriQ
--d =4p.a =-.-
4,
(13)
x
~c
r.II.e Q = -rtac4/2 ecTb o6beM }KHJI.KOCTH , nporeKaIOIUHH 3a e.II .HHHUY
speMeHH qepe3 nonepeqttoe ceqettHe Tpy6h1.
§ 25. Ilapa.11,0KC ryp6yneHTHOCTH
3KcnepHMeHT8JlbHb!e JI.8HHble B 3TOM CJiyqae B BblCilleH creneeH
3aMeqaTeJibHhJ. Xorn <JiopMyJJa (13) (3aKoH CTya3eftm1-Xaretta)
no.11.rsep>K.ll .aeTCH npH .ll .BH>KeHHH >KHJI.KOCTH a KanHJIJJHpHbIX rpy6·

§ 25. flapaaoKC TypoyMHTHOCTU
57
KaX, OHa IlOJIIIOCTblO repsier CHJIY .ll.JISI o6bllJHblX rHJJ.paBJIH'-leCKIIX
rpy6. Tol!Hee, MbI Mo11<eM c<l>OPMYJJHpoearb cJieJJ.yIOW.HH oow.Hil na-
paJJ.OKC .
n apaJI.0 Kc Typ6yJIeHTHOcTH. P.AR Te'ieHutl 8 np!l.MblX
rpy6ax 2unore3a CU.M .MeTpuu (C) U3 § 1 BbtnOAflJleTCJl, ecAU tiUCAO
PetlHOAbaca Re< 1700, u o6bt'iHO He BbtnOAHReTCR npu Re> 104 •
Ko2aa Re > 104, Ha6111oaae.Moe Ha onbtTe re'ieHue He o6Aaaaer Hu
npocrpaHCTBeHHOU, HU epe.MeHHOU CU.M.MeTpueu tt JlBAJleTCJl ryp6y-
AeHTHbt.M .
0,007
flooepxHocmHoe mpenue iJnfl oodb1 u oo:1ilyxa
0,006 +
+
• BoiJa 6 mpyoe
+
.++
+ BoJiJyx B mpy6e
0,005
+
+
••
-t
•
+ +•+
..
+
~
"i+
•++
~ 0,004
•
+++
+e.+
~
+
0,003
++
• "1.
t+
•
•
••
0,002
VD/y
•+.
I
I
I
'
2,500
10000
25000
100000
0,001
J.O 3,2
,W 3,6 J,B 4,0 4.2 4.4 4,6 4.8 5,0
Log VD/,J
Pac. 8 . no.u .o6He TeqeHH!i B03!\yxa H BO;Ibl B Tpy6ax no q11cJ1y Re.
OroeopKa «o6bltJHO» e npeJJ.bIJJ.ymeM yreep>KJJ.eHlrn orHOCHTCH
K TOMY, '-!TO MO>KHO H36e>KaTb DOSIBJJeHHSI ryp6yJJeHTHOCTH, JJ.06H-
BaSicb DOJIHOii o6reKaeMOCTH BXO.ll.IIOro oreepcTH.H, nom1pysi creHKH
H o6ecnel!HBaSI Ha exone rpy6hl JiaMHHapHoe rel!eHne. npu qpe3Bbl-
liaiiHoii rmareJibHOCTH MO)KHO 6blJIO raKHM nyreM H36e>KaTb DOSIB-
JieHHSI ryp6yJieHTHOCTH npH 3Hal!eHHSIX Re BDJJOTb .ll.O 40 000. Ho
eCJIH He DPHHHM8Tb cneurrnJJbHblX Mep, TO TetJeHHe B rpyoKax npH
Re> 2000 6y.ll .eT ryp6yJieHTHbIM.
3ro xopomo HJIJJIOcrpupyIOr KJiaccutJecKue 3KcnepHMettraJibHbie
.!l.8HHbie CraHTOHa H Il3HeJISI I)' KOTOpbie eocnpOH3Be.zr.eHbI Ha puc. 8.
') St ant on, Pan n e 11, Phil. Trans.,
A214 (1914), 119-124; 60J1ee
no.npo6Hoc 11cc.1e.a .oaaHHe ryp6y.1eHTHoro Tet1eHH11 e rpy6ax CM. e [3], rn. VIII,

58
L1. //. 17 apaaoKCbl BJl31(0i!O re11eHUJl
0HH csoeo6pa3HO no.nrnep)KJJ.aJOT ypasHeHHH Hasbe - CrnKca, no-
Ka3bIBaH, tJTO Kpuru11ecKoe 11uc;20 Pet11-10,1baca Re"P· • npH KoropoM
HMeeT MeCTO nepexo.11 . K ryp6yJieHTHOCTH, 0 .11 .HO H TO me ,UJIH 803-
Jtyxa II 80.llbl H paBHO npH6JIH3HTeJibHO 1700. TeopeTHl.feCKH 3TOT
BblBO.ll MO>l<HO 6bIJIO 6bi noJiytJHTb H3 reopeMbl 2. 60JiblllHHCTBO
cospeMeHHbIX cneuttaJIHCTOB Cl.fHTa!OT, 4TO Tel.feHHe nya3eibrn HB-
JIHeTCH npocro 1-1eycrot1ttuBblM np11 Re > Re"P · · a ryp6ynettrnoe re-
l.feHHe sce-TaKH Y.llOBJiernopHeT ypasHeHHHM Hasbe - CroKca. Xorn
H3 npHHUHna fl0.ll06HH (7) reopeMbl 2 He CJie.llyer cnpaBe,UJJHBOCTb
ypaBHeHHH Hasbe - CrnKca, HX npHrO.llHOCTb B cJiyl.fae ryp6yneHT-
Horo Tel.feHHH IlO.llTBep)K.llaeTCH OilbITHbIMH H3MepemrnMH CKOpOCTH
38TyxaHHH 0.llHOpO.[lHOH ryp6yJieHTHOCTH I).
KpoMe roro, rnnore3a (C) H3 § I sce-raKH BbinOJIHHeTCH cra-
mcru11ec1rn. 06o3H843H 4epT04K8MH cpeJJ.HHe 3H84eHHH, Mb! MO)KeM
Bbipa3HTb CHMMeTpH!O nocpe.ncrnoM CJieJlyJOW.HX cpopMyJI:
u.r=F(r), u,= u~= o. u;=0(r). ii;=H(r)
H T • .ll . TaKHM o6pa30M, paccMorpeHHe 3Kcnep1rneHT8Jlbl!blX ll8HHblX
no.ncKa3bIB8eT HaM KOHUenumo CTaTUCTU'leCK/1 onpe.a.eJiellHblX
pell!eH!III ypaBHeHHH B 48CTHblX npOH3BO.ll.HblX KaK HOBYIO H
ysneKaTeJJbHYJO o6nacrb JJ.JIH MareMaTHl.feCKHX HCCJieJJ.OBaHHH. H3y-
tJettHe T8KHX
«CTOX8CTHl.feCKHX
. ll.HcpcpepeHUH8JlbHblX ypaBHe-
HHH» OTKpbIBaeT renepb HOBble ropH30HTbl B MareMaTH'leCKOM
8H8JIH3e.
HecMOTpS! Ha .no6JJeCTHbie ycHJIHSI M8TeMaTHKOB 2). Ha6n10.nae-
M8SI HeyCTOHtJHBOCTb TetJeHHSI nya3eHJIH He nony'I aeTCSI B pe3yJibT8Te
HCCJieJJ.OB8HHH cpeJlCTB8MH MaTeMaTHtJeCKOro 8H8JIH38. Ope.JJ .nOJia-
raJIH 3) Jl.8}1{e, 'ITO B H .lle8JibHO rna.nKHX KpyrnhIX rpy6ax Te'leHHe
Oya3eHJIH SIBJISleTCH ycroU'lUBbl.M OTHOCHTeJibHO 6ecKOHe'IHO M8JlblX
B03MyUJ.eHHH. 0):{HaKo B H8CTOS11Uee speMSI .na)Ke Jl.JIH cJJy'laH .JJ .By-
MepHbJX B03MyIUeHHH cosepllleHHO Jl.OCTOBepHo ycraHOBJJeHa He-
ycTOH'IHBOCTb nnoc1<oro re1JeHHH OyaJeHJIH Me)l{.ny .n.syMH napaJI-
JieJibHhIMH nnacTHHK8MH npH Re > 5300. 0o3ToMy no.n.o6Hoe npe.n -
noJio)l{eHHe npe.n .craBJIHeTCH M8JIOBepoHTHblM.
•) Stewart R. W ., Proc. Camb. Phil. Soc., 47 (1951), 146-157. (Orno-
CHTeJJhHO H30TpOnHOH Typ6yJJeHTHOCTH CM. [14*], [9*] H npHBe,'leHHYIO T3M JIHTe-
parypy. -- flpuM. pea.)
2 ) c~1. Synge J. L. , Hydrodynamical stability. Semkentennial pub!. Am.
Math. Soc., 1 93 8, T. 2, CTp. 227-269; [36], oco6eHHO § 3.2.
3) CM. [I I). T. 2, crp. 32 -33; Como I et R.. Comptes Rendus. 226 ( 1948),
2049 (raK>Ke La Houi/le Blanche, numero special B ( 1949), 673). TeopeTHtJe·
CKHe apryMeHTbl CM. Peke r is C. L ., Proc. Nat. A cad. Sci .• USA, 34 (1948).
285-295 .

§ 26. )lpyzue napaaoKCbt cu,1u1erpuu
59
§ 26. JJ.pyrue napa)].OKCbl CHMMeTPHH
BecbMa mo6onbITHO, 4TO CKJIOHHOCTb K CHMMerp,rn np05lBJI5leTrn
JIHlllb B orpaH11qeHHOH 06J1aCTH Re< 1700. Jho6onb1rno, 'ITO H npeJJ. -
nonaraeMoe crpeMJieime K Ha11MeHbrneMy .n.eucrnmo, rro -BH .ll .HMOMy,
11Meer npHMepHo re }Ke rpaH!iUbl, nocKOJibKY pacxo.n 3Heprn11 B re-
qeHHH nya3ettJI51 MeHbllle, qeM B ryp6yneHTHOM Tel!eHHH I) .
.ll:JIH Toro lJT06bl XOTb HeMHOro pa3oopaTbC51 B 3THX qiaKTaX, pac-
CMOTpHM .npyrHe npHMepbl 113 ru.npo.n.HHaMHKH, B KOTOpblX Heoooc-
HOBaHHOe npHMeHettHe rnnoTe3bl (C) H3 § l TaK}l{e npHBO.ll .HT K He-
npaBHJlbHblM pe3yJibTaTaM.
0.n .HH H3 HHTepeCHbIX np11MepoB npe.n.craBJIHeT co6011 relJe1me
B rpy6ax c tteKpyroBb!M rronepel!Hb!M ce4em1eM. Op11 MaJibIX Re B
3TOM CJiyqae Oll51Tb-TaKH Hao.mo.n .aeTC51 rrapaJI. 'leJibHOe Te4eHHe, .L 1 Jl51
KOTOporo MOJKHO Bbl11HCJIHTb npoq:>HJib CKOpOCTett ({7], § 332) H ll
KOTOpOM npHHUHil HaHMeHbIIIero .aeucrn1rn OCTaeTC51 B CHJie. CTpH
OOJiblIIHX Re re<JeHHe CHOBa CTaHOBHTC51 rypoyJieHTHblM H .na)l{e cra-
THCTHl!eCKH He 51BJI51eTC51 napaJIJieJlbHbIM: cymecrny101 3Ha 4HTeJib-
Hhie «sropH4Hbie relJeHHH» 2) B yrnax rpy6b1.
)lpyrou cJiylJaii 6hIJI 113y4eH )l>K. TeiiJiopoM B ero KJiacc114ecKoH
paoore 3 ). PaccMorpHM BH3KYIO )l{HJI.KOCTb, HaxoJ1.HIUy10c51 MeJKJI.Y
.UB YM 51
JI.JlHHHblMH COOCHblMH UHJIHHJ.J,pa MH, KOTOpbie Bp amaIOTC51 B
npOTHBOilOJIOJKHbIX HarrpaBJieHH51X C IlOCT051HHbIMH yrJIOBbIMH CKO-
poCT51MH (J) H (J)
1
COOTBeTCTBeHHO. OrmcaHHOe Tel!eHHe SIBJI51eTCH
( rrpHOJIH}KeHHO) CHMMerpHlJHbIM OTHOCHTeJibHO rrepeHOCOB BLI.OJib II
BpameHH51 BOKpyr OCH UHJIHH.!1 .pOB, a TaKJKe He 3aBHCHT OT speMeHH.
l1Meerc51 B TOl(HOCTH 0.!I.HO peweHHe CHCTeMbl (3), (4), (6), OOJia-
. 11 .aIOmee raKoii CHMMerpuew; oHo HOCHT Ha3sau11e «TelJeHHSI Ky-
3TTa».
Opu MaJibIX (J), (J)
1
TaKoe TeI-IeHHe Kynra Ha6JIIO.UaercSI 3KCrrepH-
MeHT8JibHO. np11 OOJiblIIHX lJHCJlax PeHHOJib.D .Ca BMeCTO TelJeH!rn
Ky3TTa 110SIBJl51eTCH HeCHMMeTpH'lHOe, O. .QHaKo Heryp6yJieHTHOe re-
qeHHe. fpyoo rosopH, coxpaHHeTrn cHMMerpm1 BO speMeHH, HO He
B npocrpaHcrne.
)laJiee, paccMorpHM M aJieHhKHH ny3bipeK so3,nyxa, no.nHHMaIO-
IUHHC51 B CTOHlJeH so.ne no .n
.nef!CTBHeM COOCTBeHHOll IlJlaBylJecrn.
Bnaro.napH nosepxHOCTHOMY HaTSI}l{eHmo OH npHHHMaer cpopMy,
6JIH3KYIO K ccpepHI-IeCKOH, H , BO BCSIKOM CJiyqae, Ha Hero He .neif-
CTsyer HH o.nHa CHJia, KOTOpa51 He ObIJia Obi CHMMeTpH4HOH OTHOCH-
TeJibHO seprnKaJibHOH oc11, rrpoxo.n51mefl 4epe3 ueHrp ny3blpbKa.
Cne.11.osareJibHO, B CHJiy CHMMerp1rn, rry3bipeK .noJIJKeH Ob!JI Obi
1
) Tio noso.u .y cneu.Ha.1bHblX reopeM orHoc111e.10Ho MHHHM)'Ma pacxo.u .a 3Hep-
rHH CM. [7), § 344.
2
) [3], n. 161, me npHBO, '.\HTCSI pe3y.JJbTaTbl H!lKypa, '. l3e.
8
) Phil. Trans., A223 (1922). 289-293; 01. raioKe [36], rn. 2.

60
f A. /I, flapaaoKCbt BJIJKOZO Te<1eHUJI
IlOJI.HHMaTbCH BeprnKaJJbHO, 0)1.HaKO, H 3TO nopa3HTeJJbHbIH cpaKT,
npH Re > 50 TaKOH ny3bipeK npoKJJaJI.bIBaeT ce6e nyTb BBepx no
BepTHK8JlbHOH cnttpaJJtt! 1)
AHaJJOrHl!HOe HBJJeHHe HMeeT MeCTO B CJJe)I.e 3a KpyroBblM UH·
JJHHJI.pOM, KOTOpblH JI.BH>KeTCH B nOTOKe napaJJJJeJJbHO CBOeH o6pa-
3YIOIUeH. B JI.ttana30He l!HCeJJ PeiiHOJJbJI.ca 50 < Re < 500 3Ta 30Ha
COJI.ep>KHT qepeJI.yJOIUHeCH BHXpH npoTHBOCTOJIO>KHbIX 3HaKOB (BHX-
peBaH JI.opo>KKa Bettapa - KapMaHa); 3TO HBJJeHHe 6yJI.eT npoaHa-
JJH3HpoBaHo B § 56.
Ha nepBbIH B3rJJHJI. Mo>KeT noKa3aTbCH, l!TO no.no6Hhie npHMepbI
npornBopel!aT MeTacpH3Hl!eCKOMY rrpHHUHny Jleii6HHU.a JI.OCTaTOl!·
Horo ocHoBaHttH 2), a ttMeHHO Hameii rnnoTe3e (C). 13oJJee rJJy6oKHH
CTOJI.XOJI. COCTOIIT B TOM, l!TO, XOT.fl CU.M .MeTpUtiHble nplltiUHbl o6ycAO·
BAUBQIOT CU.MMeTpUtiHble RBAeHUR, nO'iTU CU.M.MeTpU'Utble npUtiUHbt
He o6R3aTeAbltO npuBoaRr K nO'iTU CUM.MeTpUtiHbl.M RBAeHUR.M: CHM-
MeTpHl!HaH 3a)I.aqa MO>KeT He HMeTb HH OJI.HOro YCTOU'iUBOZO CHM·
MeTpHl!HOro pemeHHH. TaKaH B03MO>KHOCTh H HBJIHeTCH .n:eHCTBH-
TeJJbHblM HCTO'IHHKOM
«rrapaJI.OKCOB CHMMeTpHH» (Ha6JJJO)I.aeMbIX
HapymeHHH rHITOTe3bl ( C) H3 § }) .
]laJJee OTMeTHM, l!TO rrpoaHaJJH3HpoBaTh ycTOU'iUBOCTb pemeHHH
ropa3JI.O TPYJI.Hee,
lleM
rroJJyl!HTb caMo pemeHHe,
rronoMy
ITO'ITH HaBepHHKa pemeHHH 6yJI.yT rroJJyl!aThCH 3a)I.OJJro JI.O
roro, KaK 6yJI.eT yCTaHorJieHa Hx tteycTOHl!HBOCTb. B cHJJY 3TOro MO-
>KHO rrpeJI.BH)I.eTh, l!TO 11 li 6yJI.yIUeM ITOTOK rrapaJI.OKCOB CHMMeTpHH
He npeKpaTHTCH.
§ 27. TeopH.H norpaHHl!HOro CJIOH
<l>yHJI.aMeHTaJJhHbIH BOrrpoc MexaHHKH >KHJI.KOCTeH COCTOHT B TOM,
lfT06bI HaHTH B3aHMOCB513h Me>KJI.Y pemeHHHMH ypaBHeHHH 3ii:JJepa
JI.JIH JI.BH>KeHHH HeB513KOH >KHJI.KOCTH H pemeHHHMH ypaBHeHHH Ha-
Bhe - CTOKca JI.JIH )KHJI.KOCTeii c 11cqe3a10me MaJioii BH3KOCTbJO. Ma-
TeMaTHtJe<:KH pel!h H)I.eT 06 aCHMITTOTHl!eCKOM rroae.neHHH pemeHHH
cHCTeMbl (3), (4) npH μ-+0 (T. e. npH Re-++ oo). CTocKOJJhKY
o6bll!HO JI.JIH Kopa6ne0 H caMOJieTOB tJHCJia PeHHOJibJI.Ca Jie:>1<aT
B HHrepaaJie 106 - 109, TO JI.JJH Toro >1<:e HHTepaana orpoMHoe npaK-
THlfecKoe 3HatJeHHe HMeeT 3aJI.atJa pactJeTa no6oaoro corrpcrrn-
BJieHHH.
1) CM. Ni s i H. and Porter A. W., Phil. Mag., 46 (1923), 754; [17],
crp. 294; Sa It man P. J ., I. Fluid Mech., l (1956), 249-275 .
2) CoepeMeHHYIO <jlopMynHpoeKy noro npHHl.lHna na.1 B i r k ho ff G. D.,
Rice Institute Pamphlet, 28 (1941), N2 I, 24-50; Collected Papers, r. 3,
crp. 778-804. Jll060nblTHO, 'ITO ¢opMaJibHble CHCTeMbl MaTeMaTH'leCKOH JIOntKH
llrHOPHPYIOT :nor npHHl.lfln, npHMeHHMOCTb KOTOporo K Qlll311Ke 6blJJa 3aMe'leHa
11 1894 r. nbepoM K10pH («Oeuvres Scientifiques», Paris, 1908, oco6e11uo
t;,Jp. 119 -215).

§ 27. Teopu11 nozpaHwmozo c.11011
61
·- --
·- ---· · ---
Jlerno mu1.eTb, '!TO o6bl1masi Teop1rn B03MymeHHH K 3TOH 3a.na4e
He npnMemtMa, TaK KaK 4JieH, Y4HTb!B310utHH BSl3KOCTb vV 2u, B ypa-
BHemm (3) u:-.1eeT caMbIH 6oJibUJoii nopHJI.OK H, c .ne .noBaTeJibHO, B03 ·
MymeHHe BH3KOCTH " OT}JOCHTeJibHO 3H84eHHSI \I = 0 eCTb CUHZYJlJlp-
1we 803Myut,eHUe 1). Tun ypaBHeHHli B 48CTHbIX npOH3BO.!I.HblX
06bI4HO onpe.neJISieTCSI 4JieHaMH HaHBb!CllJero nopsi.nKa. TaKHM o6pa-
30M, npeHe6pe>KeHHe 4JieHaMH Bb!CllJero nopSIJI.Ka ae.neT K CTHpaHHIO
pa3JIH4HH Me>K.!lY nmaMH ypaBHeHHii• .Lla*e .!I.JISI oobIKHOBeHHbIX .!I.Hcp-
cpepeHII.HaJibHbIX ypaBHeHHH T3Koro BH)I.a, KaK . _ y" + y = 0, c Kpae-
Bb!MH ycJIOBHSIMH y (0) = a, y ( l) = b, Mb! 110JiyqaeM B npe.neJie co·
BepilleHHO pa3JIH4HI,Ie KapTJ.tHbl B 38BHCHMOCTII OT roro, IlOJIO)KHTb JIH
e;-+ + 0 HJIH €-+-0.
CoBpeMeHHble HCCJie.ll .OB8HHSI YK838HH0f0 Bblllle CHHryJIHpttoro
B03MymeHHSI B 60JibllIHHCTBe HCXO.!I.SIT H3 H .!I.eH DpaHJ].TJISI O TOM ,
4TO 33BHXpeHHOCTb HMeeT MeCTO JIHlllb B TOHKOM noepaHUttHOM CJlOe
>KH.!I.KOCTH y JII06oii rnepJI.OH rpaHHl].bl, B KOTOpOM npOHCXO.!I.HT pe3-
KHH nepena.n KacaTeJibHbIX HanpmKeHHH, H B CJie.ne (48CTO 6JIH3-
KOro K BHXpeaoMy CJIOIO) 11038J].H TeJia. BHe 3TOro norpaHH4HOf0
CJIOSI H cnena Te4eHHe SIBJrneTCSI 1104TH 6e3BHXpeBbIM, H K HeMy npH-
MeHHMbl ypaBHeHHSI 3iiJiepa •
.lIJIH co6crBeHHO norpaHH4HOro CJIOSI DpaH.!I .TJlb 2) nocTpOHJI
MOJ].eJib, COrJI8CHO KOTOpoii HeKOTOpble 4JieHbl B ypaBHemrnx OT6pa-
CblB8IOTCH . .lIJIH JI.ByMepHoro IlOTOKa OH 110JIY4HJI (npeHe6perast CH·
.' IO H T S J)KeCTH) ypaBHeHHe
~
~
~1I
~
ar+u ax +vay+pP (x)=v ay2,
p=p(x), (14)
IlJIIOC 06bl4HOe YCJIOBHe HeC)KHM3eMOCTH (4) au/ax+ av/ay = 0.
YpaBHem1e (14) -ypasHeuue napa6oJZullecKoeo runa, H ero MO)K-
Ho HHTerpHpOBaTb 4HCJieHHO, IlOKa u > 0, c y4eroM KpaeBblX
ycJIOBHH u (x, 0) = v (x, 0) = 0 Ha HenOJ].BJ.l)KHOH CTeHKe u
u (x, co) = Uoc (x) BHe norpaHH4HOro CJIOH, npH4eM npeJI.nOJiaraeTCH,
11TO Uoo (x) Bbipa)KeHc, qepe3 J].aBJieHHe no ypaBHeHHIO 6epHyJIJIH:
pu;J2+ p (x) = const. Dpe.nnonaraeTCSI T8K)Ke, 4TO B nepBoti >Ke
TOl!Ke, a KOTOpoti u (x, y) < 0 npH noJIO>KHTeJibHOM y, npOHCXO.!I.HT
orpbt8 IlOTOKa 3).
6hlJIH npe.ll .npHHSITbl pa3JIHl!Hbie IlOilbITKH CJ].eJiaTb 6onee CTpo-
fHM uecKOJibKO 1rnrynTHBHbIH BbIBO.D . DpaHJI.TJieM ypaBHeHHSI ( 14);
BepOSITHO, 6onee ncero 38C.1JY)KHBaeT BHHM8HHSI TO, l!TO CJ].eJiaHO
1
) 3ro no.aqepKuea,111 Oseen ([9], crp. 211) u .ap.
2
) Proc. Third Int. Math. Congress Heidelberg (1904), crp. 484 -491,
nepe;e'l~rauo e [37].
) oo:1t!e no.!lpo6no , ·' . , t. s [3), rn. IV; 11.rn e [43]. err 94-99 11 222-224 .

62
L1. II. llapaaOKCbl Bfl3K0c0 re<teHllJI
MH3ecoM 1). O.nttaKo MHorne pe3yJihT3ThI Teop1m ew.e
.u aJieKo tte
SICHhI. KpoMe Hepa3'hSICHeHHOH CHHryJrnp11ocTH y nepe!l.Heii KpOMKH,
K HHM OTHOCHTCSJ TaK)l{e CJ1e.uy10lll,HH napa.ll .OKC .
napa.uo Kc norpaHH4Horo cJIoSJ. Teoperu'lec1-.u
npu
BbtBooe ypaB1-1e1-111R ( 14) npeanoAa2aerCR, 'lTO ornoutenue a/x ro.1-
w,unbt no2pa1tU'lHOW C,10R a K OALLHe x crpeMUTCR K HYAIO. 3Kcne-
puMeHTaAbHO ,ice nOKQ3at-to, •tro eCAU a;x < 0,01, TO no2paHU'lHblU
CAOU CTQHOBUTCR ryp6y.,zeHTHblM U ypaBfteHUe ( 14) He yiJoaAeTBO·
pReTCR 2 ).
HanpHMep, norpaHH'IHbIH cJioii ocraerc51 JiaMuHapHbIM B!l.OJih
Kopnyca Kopa6JIH H B.ll.OJih KpbIJibeB caMoJieTa so apeMH noJieTa
Bcero Ha pacCTOSJHHli HeCKOJlbKHX caHTHMeTpOB! 3ry Typ6yJieHT·
HOCTb norpaHH4HOro CJIOH MO)l{I-IO CB513aTb, KaK Mb! ceiI'laC YB!!·
lUIM, c Typ6yJieHTHOCTb!O s Tpy6ax.
§ 28. napa.uoKcbl 3A«IJeJisi H Aio6ya
npe!l.CTaBJieHHe O TOM, 4TO conpoTHBJ1em1e rnapH.LI.a D .IJ.OJDKHO
6b1Th nJJaBHo Bo3paCTa10w,eii cflYHKUHeii CKopocrn ettapSJ.ua v, secbMa
crapo. TaK, BO MHOrHX y4e6HHK8X MO)KHO H3HTH «.ll .OKa3aTeJib-
CTB8» ( c noMOlll,blO attaJIH3a pa3MepHOCTeH, CM. § 61) TOro, 4TO CO ·
npOTHBJleHHe D .ll .OJl>KHO 6bITb nponopUHOHaJlbHO v nptt MaJlblX
CKOpOCTHX H nponopUHOHaJlhHO v 2 npH 60JJbllIHX CKOpOCTSJX. 003TO-
MY B BhICUJeH cTeneHH Y.ll.HBHTeJJbHbIM noKa3aJJcSJ OTKpbIThlH B 1912 r.
KoHcraHUH H 3iicpeneM 3) cne.uyJOll.lHH napa.uoKc.
O a pa .lJ. 0 KC 3 ii cfl e JI SJ. Ilpu 'lllCAQX Pet1HOAb8ca, 6All3KUX
K Kpuru'leCKOMy i;.ucAy Re"P· ~ 150 000, conporuBAeHue crpepb,
rpaKruttecKu y6brnaer c 803pacra1weM cKopocru.
.IlBa ro.ua cnycrn CTpaH.UTJJh noKa3an, 4TO 3TO na.uettHe conpo-
THBJieHHSI 38BHCHT OT B03HHKHOBeHHSJ ryp6yAeHTHOCTU B «norpa-
HH4HOM CJIOe» OKOJJO cqiepbl H 3Ta Typ6yJieHTHOCTb MO}KeT 6b1Tb
Bbl3B8Ha nyTeM yBeJJH4eHH51 UiepoXOB3TOCTH cqiepw UAU }Ke npH
nOMOIUH .ll.OHOJlHHTeJlbHOH Typ6yJIH33UH!i HOTOKa . .lI.eHCTBHTeJlbHO,
1) ZAMM, 7 (1927), 425-427 .
.9epHCTHlfecKoe Hcr.~ e.1toeam1e CHHrym1pHOCTH
.1 1.Jlll TelfeHHSI e6JJH311 nepe.1tHeH KpOMKH flJlOCKOH nJJaCTHHKH CM. c a r r i e r G. F .,
Lin C. C ., Quar. Appl. Math., 6 (1948), 63-68 .
2) Cp. [43), crp. 32 -33 .
,l lpyrHe TPYllHOCTII, B03Hl!Ka!OU!He npll H3HBHOM
npuMeHeHHij erecp1111, ynoMsrnyn,1 raM 11a crp. 110-112. no noBOllV J1aJ1hHeuw11x
3arpy.1t11e1t.!IH CM . TaK>Ke Stewarts on K.. I. Math. Phys., MIT. 36 ( 1957),
173-191 .
3)Eiffel G.,
Comptes Rendus, 155 (1921). 1597-1599 . 06·b11cHem1e
npaHJ1rnsi CM. e [11), r. 2, n. 63, 11.111 Gott. Nachr. , Math.- Phys. K,1. (1914),
177-190,

§ 29. Pei!y.zupOBQHlll! 110i!f)Qllll'IH020 CAOJl
---------------·--··---··--·-·--·--·-·· ·-----
MO>KHO CBH3aTb ti11cJ10 Re 1:p. --=: 150 000 napa.iJ.OKca 3it<peJI.st c
ReKp. ~ 1700 JlJUI ryp6yJ1enrnocm n 1py6ax nocpe.u .craoM conocra-
BJiemrn TOJIIUl!Hbl JiaM1rnapnoro norpam11rnoro cJioH a.D .HaMeTpy Tpy-
6bI d. ECJIH Re = ucox/v ecrh llHCJIO PeifHOJib.D .Ca ocHOBHoro TetJe-
HHH Ha pacCTOHHHH x OT nepe,llHeH KpOMKH, TO H3 ypaBHeHHH (14)
cJie.llyer a(x) ~ 4 Vvx/tl,:x,, 11 ror,lla lIHCJio PeHHOJib.llCa JlJIH norpa-
HHtJHoro CJIOSI onpe}leJISieTCSI Bbipa:>KeHHeM Reo = u,;;oa/v ~ 4 Vucoxfv.
Orcro.Qa o/x ~ 4/VRe, a •rnc.110 PeHHOJib.D .Ca HJ napaJJ.OKca 3ii-
<f>eJISI Re.lip .!::":: '. 150 000 npHOJIH:>KeHHO cooTBercTByer l!HCJIY Rea~
~4 V ReKp. :::::'.1600, l!TO anoJIHe cornacyercSI c Re1,p. JlJIH ryp6y-
JieHrHocrn B rpy6ax. 8TO OTKpbITUe 06'bSICHSleT T3K:>Ke CJie,llyIOIUHH
6onee crapbIH rrapa)J.oKc.
n a p a .11 o Kc .IT. IO 6 y a. Conporae1teHue na1tKU, ,coropy,o yiJep-
o,cuea,or nenoiJB.UOICHO e noroKe, u.MelOU{eM cKopocTb v, o6blttHo
MeHbtll.e, tteM conporuB,teHlle rot1 3/Ce nallKU, ,coropy,o TRHYT c rofl
rHCe CKOpOCTblO v B CTORtteii BoiJe ..
3TOT rrapa.noKC ocooeHHO JtHTepeceH IlOTOMY, l!TO Ha rrepBblli
B3rJISIJl Ka:>KeTCSI, 6yJlTO OH npOTIIBOpetmr OCHOBHOMY npHHUHny
MexaHHKH HbIOTOHa - mmapnaHTHOCTH Bcex 38KOHOB rrpu nepexo,!J,e
K paBHOMepHO H nocrynareJibHO ABH}f(YlUHMCSI OCSIM KOOP.!I.HHaT.
BepoSITHO, noroMy qro JleoHap.11.0
.u .a BHHllH 1) npH3HaBaJI aror
npHHU.Hil, OH yTBep:>K.l{aJI paBHOCHJlbHOCTb ABYX yKa3aHHblX BblWe
cJiyqaea - xorSI aro cpa3y onpoaepraercH Ha6JIIO.UeHHeM.
B H3CTOHmee BpeMH o6'hSICHeHHe napa.u.oKca .IT.ro6ya ClJHTaeTCH
H3BecTHbIM. noTOKH :>KH.UKOCTH Beer.LI.a 6onee HJIH MeHee ryp6y-
JieHTHbI; 3TO npHBO.U.HT K nOHH:>KeHHIO conpoTHBJieHHSI no TOH :>Ke
(He o6'bHCHeHHOH MaTeMaTHtieCKH) npHl!Jrne, no KOTopou IlOHH-
:>KaeTCH conpOTHBJieHHe npu o6reKaHHH c<t>epbI, KaK 6bIJIO noKa3aHo
npaH.U.TJieM. Bbipa>KaHCb COBpeMeflHbIM S13blKOM, CB060.ll.H3SI ryp6y-
JieHTHOCTb noTOKa Bbl3bIBaer nepeXO.ll. K ryp6yJieHTHOMy .ll .BHlKeHHIO
B norpaHHlJHOM CJIOe. 3TO B CBOIO oqepe.u.b 38.U.eplKHBaer orpbIB no-
TOKa, cy>KaSI T8KHM o6pa30M «cne.u .» H yMeHbWaSI CBH3aHHOe c 9THM
no6oaoe conpornsnem1e.
§ 29. Perynnpoaauue norpaautJHoro CJIOil
nepBOH3llaJibHO ( CM. npHM. 1) Ha CTp. 61) npaHJJ.TJib CllHTaJI
CBOIO reopHIO norpaHH'IIIOfO CJIO.SI MOCTOM, CB.Sl3bIB8IOIUHM KJl8CCH·
qecKyro reoperuqecKyro re.u.poJJ.HHaMHKy n .u.nHaMHKY peaJibHbIX
1) Mac Curdy E., The Notebooks of Leonardo da Vinci, New York,
1941. CTp. 503 . Jleotrap~o T3K,KC 33MeTH. 'I
OO)lOOHe B03Ayxa H 80,l(bl (raM )Ke.
CTp. 645).

64
f A. // . llapadDKCbl BR3KOZO Te11eHUR
>KH.llKOCTeH. CornacHO TeopHH, npH HeorpaHH'- leHHOM BOJpacTaHIIII
'I H C Jia PeAHOJTbJICa Re norpaHH'lHblH cnoH JIOn>KeH crpeMHTbCH cTaTb
6ecKOHellHO TOHKHM. 0TCIO .ll8 cne.11yeT,
'lTO npH JlOCTaTO'lHOH
HJo6peTarenbHOCTH Mbl 8 COCTOHHHH pezyAupOBQTb ::nor norpaHH'l·
HblH CJTOH npoHJBOJlbHO MaJiblM ycHnHeM TaK, ' IT 05bl annpOKCHMHpO-
BaTb re'leHHH )l(yKOBCKOro c HyAeBblM conpOTllBAeHUeM u 6oAbtUOU
noa1ieMHOU CUAOU.
.llm1 JlOCTH>KeHHSI 3TOH uenH 5hIJTO npoee)leHO
orpOMl!Oe 'lHCJIO Tlll.aTeJibHblX HCCJle)lOBaHHH H 3KCnepHMeHros; He-
KOTOpble HJ HHX Mbl ceifqac BKparue paCCMOTpHM I).
HaH6onee HJBeCTHblM HBJJHerc.11 TO coo6pa>KeHHe, 'lTO conpoTH-
sneHHe MO>KHO yMeHblllHTb npH nOMOlll.H ynyqllJeHHSI 061e,caeMOCTU.
no.LI. 3THM Mbl noHHMae\1 no.D .6op .llJIH Kpblna ca Monera raKoro oqep-
raHHSI, KOTOpoe CBO.[I.HT K MHHHMYMY rpa.D .HeHT npoTHBO.ll3BJTeHHH.
3TO .llOJI>KHo 6b1no 5hl Ja.D .ep>Karb orpble noToKa H TaKHM o6pa30M
yMeHbllJHTb no6osoe conporHBJieHHe, noJBOJTSIH HJ5e>KaTb JaCTOHHOH
o6nacrH y nepe.D.HeH KpOMKH JIH6o oTpblea e6JIHJH Hee. Ha npaK-
THKe 3TO .lJ.OCTlfraercSI TeM,
'lTO nepe.D.HSIH KPOMKa oxpyrnsieTCH,
a ocranbHaSI 'lacTb rena nocreneHHO cy>1<1-1eaercsi JIO ocrpoH Ja]lHe11
KpOMKH.
OnacHOCTb nosieneHHSI JacTOHHOH o6nacrH, xoropall yMeHbUiaeT
noJI"ber.rny10 c1rny, paeHo KaK yeenH•rneaer no6oeoe conpornene-
1rne, npe>1<.ne ecero BOJHHKaer npH 6oJJblllHX yrnax araKH .
.l lnsi Toro 'lT06bl Ja.D.ep>K3Tb nOSIBJieHHe 33CTOHHOH o6nacTH, BeCbM3
noneJHO TaK>Ke cnerKa HCKpHBHTb npotpHJTb Kpblna KHHJY, B npe-
. ne nbHOM cny'lae npotpHJJH e BH.lle .llYrH oKpy>KHOCTH nerKo y6e-
. llHThCSI B TOM, ' lTO 3TOT npHeM nOJBOJISleT HJOe>KaTb 6eCKOHe'lHOrO
3Ha'leHHSI CKOpocrn H3 nepe.D .Hefi KpOMKe; B 06lll.eM cny'lae Te'leHHH
)l(yKOBCKOro 3TO npHBO.llHT K 3H8'1HTeJibHOMY yMeHbllJeHHIO rpa.D.H ·
eHTa npOTHBO}laeneHHSI Ha eepxHeH (nO.llCOCHOH) CTOpoHe.
rnaettoii JaJia' - leH HaUHOHanbHhlX na6opaTopHH e Te'- leHHe
1910-1930 rr. 6blJIO COJ):laHHe onTHMaJTbHO o6TeKaeMblX 04ep-
T3HHH nnsi caMoneroe, neraeurnx co CKopocTSIMH He 6onee
400 ICM/~ac (Karna MO>KHO 6bJJIO npeHe6pe'lb 3<Jlq>eKT3MH C>KH·
MaeMOCTH) .
.l lnsi aHa~H3a 3KcnepHMeHTaJlbHblX .ll3HHblX H npoeKTHpoeaHHSI
11oeblx onhlTHhlX KOHCTPYKUHH 6onbll!HM no.D.cnopbeM 6bln pac'-leT
pacnpe.D.eJieHHSI .D .aeneHHH cornactto TeopHH )l(yxoecxoro, a cne .D .o -
uaTenhHO, no ypaeHeHHSIM 3Hnepa. 0 .D .HaKo ueHHOCTb TaKHX pac-
' leTOB He e onpe.D .eJieHHH JHa'leHHH nO.ll 'beMHOH CHJTbl, no6oeoro CO·
npoTHBJleHH.SI HJTH MoMeHTa (cp. § 8), a e TOM, 'lTO OHH noJBOJTHJIH
yKaJaTb Ha nepeXO.ll K ryp6yneHTHOCTH H Ha orpblB nOTOKa B no-
1) CM. [3), rJT. 12 H [43], rn. 13, r.11e co,!1 .rpit<IITCR 60.,ee no.woe eee .11eHHe
B paccMaTpHeaeMblil eonpoc. 0THOCHTeJlbHO 11.11 .eil npaH.11TJIR CM. [11), T. 2,
n. 50-52 II 9 l-9J,

I 29. PuyAuposa11ue no2pa11urtH020 CAO.II
65
rpaHH'IHOM cJioe. KaK 3TO JJ.aJieKo OT nepsoua'laJihHOH KouuenuHH
Jlarpam1rn!
CTpH 60JibWHX yrnax aTaKH MO:>KHO H36e:>KaTb rrorepH CKOpocrn
u noJiy'IHTh 6oJibllIYIO IIOJJ.'beMHYIO CHJIY npH noMOlllH raK Ha3hmae-
MhIX pa3pe3H.blX KpblAbe8. TaKHe KpbIJibSI 3HaKOMbI nacca>KtipaM
caMoJieTOB H MoryT ObITh noJiy'leHbI npH IlOMOJ.l{H npeJJ.Kpb!JIKa H _3a-
Kpb!Jll<a. K co:>KaJieHHIO, paape3Hhie KphIJibSI yseJIH'IHBaIOT Jio6oI3oe
conponrnJieHHe, Il03TOMY HX HCilOJib3YIOT JIHlllb Ha B3Jiete H rrpH
nocaJJ.Ke, KOrJJ.a B nepsy10 oqepeJJ.h B8}1{HO noJiy'IHTh 6oJibUIYIO IIOJJ. ' b ·
eMHYIO c1rny npu yMeHhllleHHOH cKopocni:. Xorn TPYJJ.HO rrpeJJ.cKa-
38Th MaTeMaTH'leCKH, KaK pa6oTaIOT paape3Hble Kpb!JlhH, xapaK-
rep BJIHHHHSI meJieH Ha re'leHHSI BJJ.OJib sepxHeH CTOpOHbl Kpb!JibeB,
O'leBHJJ.HO, rro.uo6eH JJ.eikTBHIO crpyii, KOTOpbie CHlUKaIOr TeHJJ.eH-
UHIO K OTpblBY IIOTOKa nocpeJJ.CTBOM ycKopeHHSI norpaHH'IHOro CJIOSI.
lfao6peTaTeJihHhie reXHHKH npo6osaJIH TaK:>Ke HcnOJih3oBaTb crpyH
JJ.JIH rex me ueJieii.
llpyroe MHoroo6ema10mee rrpHcnoco6JieHHe ocHoBaHo Ha coa-
JJ.8HHH rrpHHYJJ.11TeJihHoro noacoca JIHoo qepe3 meJIH, JIHoo qepe3
paBHOMeptto pa3MemeHHhie Kpyrnhie OTBepcTHSI Ha Tex yqacrKax,
rJJ.e 1rna tJe npor1aollleJI 6b1 orphIB norpaHH'IHOro cJiosi. B 3TOM cJiy-
qae rrorpaHHlJHbIH CJIOH OT}KHMaercsi K CTeHKe, H Mbl OIISITb IIOJiy-
tJaeM JIY'llllee rrpH6JIH>KeHne K re'leHHIO )KyKoBcKoro. EcJIH 11cnoJib-
ay10rcsi meJIH, TO, HCXOJJ.H H3 reopm-1 )KyKOBCKOro, HY:lKHO C03JJ.8Tb
IIOBhIUieHHoe JI.8BJiem1e KaK pa3 snepeJI.H meJieii 1). Mo>KHO TaK:>Ke
IIOilbITaThCH HCIIOJib30BaTb IlOJJ.COC JJ .JISI Toro, 'ITOObl coxpaHHTb no-
rpaHH'IHhIH CJIOH AQ.MUH .apH.bl .M ., TeM caMbIM OilHTh·TaKH yMeHbUiaSI
JI06osoe conporHBJieHne. K co:>KaJieHHIO, oqeHb TPY.llHO, no-BHJJ.H ·
MO My, IIOJIYtJHTb TaKoe JI8MHHapHoe re'leHHe. .lI,a:lKe JieTSilllHe B
B03J.lyxe HaCeKOMbie Moryr Bbl3Barb ryp6yJieHTHOCTh npH o6reKaHHH
caMoii rnaJJ.KOH rroaepxHOCTH KpbIJia.
CTocJieJJ.HSISI HJJ .eSI CTpaHJJ.TJISI 38KJIIOtJaJiaCb B TOM, 'ITOObI IlOMe·
lll8Tb yMeHbllleHHIO CKOpOCTH B norpaHHlJHOM CJIOe, a CJieJJ.OBareJibHO,
H orphrny, HCIIOJih3YSI oeu:»CyU{uecR zpanuu,bt. Xorsi B na6oparopHbIX
ycJIOBHSIX H MO:>KHO npoJJ.eMOHCTpHpOB8Th npaBHJihHOCTh paccMa-
TpHaaeMOH HJJ.eH 2), HO JJ .O CHX nop ee np1-1MeHeHHe HMeer JIHlllh
3MnupuqecKylO ocHoay, raK tJTO B JJ.aJihHeiirneM MbI 60J1hrne He 6y-
.u.eM B03BpamarbcH K 3ToMy aonpocy.
1) CM. Goldstein S., J. Aer. Sci., 15 (1948), 189-220; Pfennin-
ger W., T3M >Ke, 16 (1949); 227-236; Doehnhoff A. E ., Lofton L. I<.,
Jr., TaM )Ke, 729-740; Lach man n G. V ., J. Roy. Aer. Soc., 59 ( 1955),
163-198.
2 ) CM. [11]. n . 51 -52; Acker et J., Das Rotorschiff..., Goettingen, 1925;
Fa v t e A .. Comptes Rendus, 202 (1936), 434-436. no noBOAY u.11.eii IlpaHJJ.TJiff
CM. TaK>Ke § 9.

66
f A. II. llapadoKCbl BR3K020 Tt'ltHUA
TexuuqecKHe TPY.D .HOCTH, BCTpeqa10muecsi npu peaJJH3aQHH yno-
MHHyTl:,IX BblWe coo6palKeHHH, He JJ.OJilKHbl 38CJIOHS!Tb Jie:>JCalUylO
B HX OCHOBe H.D.elO - annpoKCHMa)lHIO H)J.eaJlbHOro Te'!eHHH )KyKoB-
CKOfO, onucaHHOro a § 8.
§ ao. napa,11.oKc CToKca
B § 25-29 Mbl paccMOTpeJJH TPYJJ.HOCTH, CBS!38HHble c TeopeTH-
qecKHMH pacqeTaMH Teqeuuii npH 6oJiblliHX qucnax Re. Tenepb Mbl
nepeii.D.eM K npo1usonoJio:>JCHOMY cnyqa10, KOrJJ.a Re-0. B 3TOM
cnyqae pa3JJOJKeuue no CTeneusiM Re ylKe ue CBH3auo c «cuurynstp-
HblM B03MymeuHeM> B CMblCJie § 24; HeJIHHeHHbIH KOHBeKTHBHblH
qJieH u. Vu He t'. iyJJ.eT qJJeHOM caMoro Bb!COKOro nopHJJ.Ka, H c Ma-
TeMaTHqecKOH TQqKH speHHS! npe,ACTaBJIHeTCH BOOJIHe QeJiecoo6pa3·
HblM ero nonpocTy onycTHTb.
3ro 6blJIO CJJ.eJJ3HO CTOKCOM, KOTOphlii BBeJJ, TaKHM 06pa30M,
HOBblA KJiacc HJJ.eaJihHblX ret1euuii, o6blqno ua3blsaeMwx «non3y-
ll\HM1t>. B TaKOM npH6JJHlKeHHH C10Kc BblBeJI lf>opMyny
D=61t11CV
(15)
.n nsi conpoTHBJieHHH, HCnblTblBaeMoro TBepJJ.oii clf>epoA pa.a .uyca c
npH Me.l{JieHHOM JJ.BHlKeHHH co CKOpOCTblO V B >KHJJ.KOCTH C BH3KO·
CTblO μ. Ilpu BhlBOJJ.e cymeCTBeHHO HCDOJlb3YeTCSt runore3a (C) npH-
MeHHTeJibHO K ocesoA CHMMeTpHH. 0KOHqaTeJJbH8St (f>opMyJJa (] 5)
xoporno noJJ.TBep:>tCJJ.aeTcsi 3KonepHMeHTaJI&Ho 1) npu Re< 0,2.
l(a3aJJOCb 6bl, BOOJJHe ecTecTBeHHO npHMeHHTb TOT JKe caMblH
MeTOJJ. K KpyrDBblM UHJIHHJ1.P8M, J1.BHlKYll\HMCSt nepneH.ll .HKyJJS1puo OCH.
0.anaKO B 3TOM CJiyqae Mbl HMeeM CJie,aylOlUHH napaJJ.OKC.
n a p a .a o Kc CT o Kc a. CraijUOHapHoe moA3yw,ee, 061e,ca·
HUe ,cpy20BOW l{UAUHapa HeB03MO:JICHO.
Jl. 0 Ka 3 a Te JI b CT 8 o. Ilpu [lJIOCKOM CTaQHOHapHOM [l0JI3yll(eM
reqeHHH ypaBHeHHSt Has&e - CTOKCa (]]) CBOJJ.HTCSI K ypaBHeHHIO
V2C = O. EcJIH V - lf>YHKI.\H.H ToKa, TO nocJieJJ.nee ypasueuue 3KBH -
BaJieHTHO ypaBHeHHIO V4 v = 0, T. e . v - 6u2apMOHU'(eC1CQJl (f>yHK·
QHSI. 0TCIOJ1.a CJie.ayer, 'ITO v - QHQAUTUltec,ca,i <pyHKijUJl 2). Jl.eii·
CTBHTeJlbHO, BO BCHKOM KpyrosoM KOJlhlle lf>YHKQHIO V MOlKHO pa3·
JIO:>KHTb B psi.a. cl>yp&e
V = ~an (r)cosn6+bn (r) sinn6,
(16)
1
) [3J, n 215. no OOBOAY 4>opwynbl (15) CM. (7}, n. 337-338.
.
2) 0THOCHTeJibHO CBOACTB Hcnon1,3yeMblX 6HrapMOHH'leCKHX <l>YHKttHA CM.
NI co I es cu M., Les fonctions polyharmoniques, Hermann, 1936, ocotSeHHo
crp. 13-16 H crp. 32. 06bl 'IHO BMecro crpororo .11 .oxa3aren&eraa ccblJ18JOTCJ1 Ha
rHnore3y (E).

§ 30. flapado,cc Cro,,ca
67
r.11 .e Un (r) H bn (r) yJI.OBJiernops110T COOTHOllleHHS!M
(16')
EcJIH BeKTOp CKOpOCTH (oV/oy, -
oV/ox) B npsiMoyrOJihHhlX KO-
Op.ll.HH8T8X orpaHH'leH Ha 6eCKOHe'lHOCTH, TO npocToe Bh!'lHCJieHHe
)I.aeT coOTHOlllem1e a 1 (r) = A,r + aifr, H an (r) = a~/r~ -
2 + a:;,n
npH n >2. 0TC!OJI .a cJie.ll.yeT cpopMyJia
V- +'?+ I +a1cosB+b1sin6 +
_
vy
v0
a0 nr
r
co {(
,
")
(b'
")
}
+~ ,:~2+:: cosnB+ ,n~2+:: sinnB .
(17)
2
c .npyroi"1 CTOpOHbl, CCJIH V4v = 0, TO 113 TeopeMbl O JI.HBepreHUHH
cne.11 .yeT ypaBHemt:e
ff(V2V)2dxdy = f{v2V
0
0~
-
v-:n (V2V)} ds.
A
C
r.11.e C - KpHBaH, orpamt:'lHBa10maH o6nacTh A.
nycTb Tenepb A - o6JiaCTb Me)l{.ny UHJIHHJl.POM H 60JiblllOH
OKPY>KHOCTblO pa.11 .Hyca r. TaK KaK Ha UHJIHHJ].pe v = av;an = 0, HH-
'ferpan B npaBofl II8CTH npe,nhI.ll .Ylll.ero ypaBHeHHH, B3HThlH no 3TOH
qacTH C, .non>KeH o6pamaThCH B HYJib. Ha 6oJihlllOH oKpy>KHOCTH,
IlOCKOJihKY
cnpaBeJI.JIHBO COOTHOlllCHHe
f{V2V ~: - V :n(V2V)}ds= 0 (,-2).O(r)~O
c
npH ,._. o o. TaK KaK (V2V)2 > 0, TO 0Tc10.11 .a cne.nyeT, 11To
V2V === 0, T. e . v Jl.OJI>KHO 6h!Th rapMOHH'leCKOH cpyHKUHeH. Cne.no -
BaTeJihHO, B cpopMyJie ( 17) a:= b: = 0. HaKOHeu, H3 ycJIOBHSI
av/ar = 0- ycJIOBHH npHJIHnaHHH Ha IlOBepXHOCTH UHJIHHJl.pa -
cne.nyeT, 'ITO v = Vo H tJ = 0, a 3TO 3aBeplllaeT Jl.OKa3aTeJihCTBO •) .
1) no)J.poCittoe 1131101Kem1e eonpocoe § 30 H § 31 CM. a [l*}. -flpu.41, pea.

68
I'. 1 1. // . flapaaotcCbl tJR3KOi!O re11eHU,i
§ 31. YpaaeeeHfl OseeHa
Q3eeH 1 ) H .naM6 BBeJIH napaJJ.OKC CTOKCa B paMKH reopHH, no-
Ka3aB, 'ITO KOHBeKTHBHhle 'IJieHl>I npeo6na.n .a10r HaJJ. BH3KHM 'IJieHOM
npH 011euh 6oJihUIHX aHa11eunsix ,, K8K Ohl HH 6hIJio Mano 11ncno Re.
OepeonpeJJ.eJieHHOCTH MO}KHO H36e}K8Th, 6onee aKKypaTHO nepeXO)lH
K JJ.BOHHOMY npe.a.eny npn Re-+ 0, r-+ + oo.
lITOOhl nony11HTh pa3peUIHMYJO KpaeayIO 3a.n .a11y, Q3eeu
npe.n .JIO:>KHJI aaecrn B oneparop D/Dt BMeCTO TO'IHhIX 11neuoa
~ u1,.a/ax1,. JlHHea pH30BaHHble KOHBeKTHBHble 'IJieHbl ~ Uk (00) a/axk.
oJiaro)lapsi BBe.n .eHHIO raKHX cnaraeMhIX B ypaaueum1 CroKca 03eeH
cMor nonyttHTh reopern11ecKyIO <f>opMyny .a .nsi no6oaoro conporHBJie-
HHH B cny11ae Me.n.neHHO JJ.BH}Kyl.l .{erocsi 11.HJIHHJJ.pa. npu6JIH}KeHHOe
3KcnepHMeHT8JlhHOe IlOJJ.TBep>KJJ.eHHe 9TOH qJOPMYJihl BO:i3MO>KHO,
XOTH H OK83hIBaeTCSJ JJ.OBOJlhHO TPYJJ.Hh!M ((3], rJI. IX).
3To paapeUieHne napa.n.oKca CTOKca B ceoIO 011epe.n .h npHBeJio
K .npyroMy napa.11.oKcy, OTKPhITOMY cl>aifnouoM 2). B napa.n .oKce cl>aii-
JIOHa yrnep>K.naeTCH, 'ITO ypaBHeHHH Oaeeua, B3S!Thle 6yKB8JlhHO,
)laIOT 6ectwHettHblU MOMeHT JJ.JIH 3JIJIHDTHlleCKOro UHJIHH.llpa, KOCO
DOCT8BJieHHOro OTHOCHTeJihHO IlOTOKa. 3TOT napa.n.oKC 6hlJI ue.naBHO
pa3peUieH I1MaH npH IlOMOl.l.{H nepexo)la K 6onee BhlCOKHM npHOJIH·
>KeHHHM.
0pH6JIH>KeHHhle ypaaueHHSI Q3eeHa MO>KHO TaK}Ke HCil0Jlb30B8Tb
,D .JIH ncnpaaneHHH qiopMyJihl ( 15), 11To6hI y11ecTh BJIHHHHe MaJioro,
uo Koue11uoro 11ncna Re ua no6oeoe conpoTHBJieune c<f>ephl; nonpa-
BO'IHhIH MHO>KHTeJih oKa3ancsi paaHhlM (I + 3Re/8). 3roT nonpa-
BO'IHhIH MHO>KHTeJlh 6blJI Tlll.aTeJihHO HCCJie.n.oeau foJibJ).UITeHHOM 3)'
KOTOpblH DOJIYLIHJI CTeneHHOH pHJJ. JJ .JISI K03qJqJHU.HeHTa conpoTHBJie·
HHH Cv(Re), cxo .n.Hl.l.{HHCH, sepoHrno, npn Re< 2. 3KcnepHMeHT8Jib-
Hble H3Mepemrn, no-BHJJ .HMOMY , JJ.aIOT MeHhlllee conpoTHBJieune;
KpOMe Toro, BBHJJ .y aCHMDTOTH'leCKOro xapaKrepa HCCJie)loBaHHH
Q3eeHa B03HHKaeT Bonpoc, He 6yJJ.eT JIH OKOH'laTeJibHaH cpopMyJia
sepHa TOJlhKO 8CHMDTOTHlleCKH 4).
06aop pemeHHH .llpyrnx KpaeBhIX 3a.na11, K KOTOPhIM npuBO.llHT
ypaBHeHHSI OaeeHa, Jl.8H B pa6oTe [9], '18CTh III. 0.nHaKo annpoK-
CHM8UHH KOHBeKTHBHblX 'IJieHOB secbM8 HeTO'IH8 BOJIH3H npenHT·
1) [9), cTp. 162; [7], CTp. 769; CM. TaKiKe [12], T. VI, CTp. 29 -40 .
2 ) Fil on L., Proc. Rog. Soc., All3 (1926), 7-27 . no noBO.llY 061,11cHeHHH
napa.llOKCa cJ>aAJJOHa, JtaHHOrO 11MaH (Imai), CM . TaM iKe, A208 (1951).
487-516 .
3) GoIdstei n S.,
Proc. Rog. Soc., Al23 (1929), 225-235; HJIH [3],
§ 2f5. Cp. Wey s sen ho ff J., Annalen der Physik, 62 ( 1920), 1-45 .
4) C~·.
I(aplun S., Lagerstrom P.A ., 1. Math. Mech., 6 (1957),
585-593; Proud man !., Pe arson J. R. A.. /. Fluid 1,-tech., 2 (1957),
237-262.
.
.
..

§ 32. napaaoKC 11!{3blpbKa
69
CTBHH H CTeHOK. B CBHaH C ~THM reopHH norpaHHl!HOro CJIOH OpaHJl-
TJI.SI npH DOJihumx l!Hc.nax PeiiHOJih,nca 3Ha' -I HTeJibHO 6onee nJIOJJ.O ·
TBOpHa.
§ 32. napa.11.oKC DY3hlpLKa
B reopHH IIO,D;BOJJ:HOro B3pbrna Mb! BCTpeqaeMCH C IlOJIO>KEHHeM,
aHaJIOrHl!HbIM napa.noKcy CrnKca. Xorn cyll.l ,ecTByeT npocraH H
qpe3Bb!ll8HHO IlOJie3HaH reop1rn ccpepHl!eCKHX rry3blpbKOB, B03HH-
K8!0ll.l,HX npH llOJJ:BO.UHbIX B3pbrnax 1), JierKo Il0K838Tb, l!TO B .uay-
MepHOH rn.upo.n1rnaMHKe JJ:JIH BCHKOro pacIIrnpeHHH HJIH C)Kanrn
IlY3blpbKa B HeC}l{HMaeMOH >KHJJ:KOCTH rpe6yeTCH 6ecKOHel!HOe 3Ha-
qeHHe KHHeTl-!l!eCKOH 3Heprm1
~ff (p'vU · 'vU)dxdy.
CT pH KOHe11HhlX c1rnax C>KHMaeMOCTh Bcer.n:a
. n: om1rna Hrpa rh
OCHOBHYIO pOJib Ha .n:ocTaTO'IHO DOJlbllIHX paCCTOHHHHX.
HMeIOTCH ell.l ,e
. D.Ba JII06onbITHbIX rrapa.n:oKca, rrpoucxo>1<.n:em1e
KOTOpblX CKopee !pH3JF!eCKOe, He)KeJHI MareMarn:qecKoe, H KOTOpble
IlOK83blB8IOT YH3BHMOCTb rHilOTe3bl (A) H3 § 1. nycTb MaJieHbKHH
B03.UYIIIHbIH rry3bipeK rro.UHHMaerrn B )KHJJ:KOCTH rro.n:
.u eiicTBHeM
IlJiaByl!eCnI, rrpHt.1eM OH HaCTOJihKO MaJI, l!TO BCJieJJ:CTBHe rroBepx-
HOCTHOro Harn>KeHJrn coxpaHHeT 110'-I T H ccpepHl!eCKYIO cpopMy 11 era
JJ:BH>KeHHe -
«IlOJI3Yil!ee». TaK KaK rry3bipeK COCTOHT H3 ra3a, TO
BMeCTO ycJIOBHH (6) Ha.no B:HITb ero JIOrHl!eCKOe 0606mem1e: u (x)
,n : oJI>KHa DbITb Herrpepbrn11a npH nepexo.ne Liepe3 rpam1uy.
CTocraaJieHH8H ,aKHM o6pa30M MareMaTH' -lecKaH 3a.nat.1a DhIJia
peIIIeHa PbID'- I HH CK H M H
A.naMapoM rrpH .no6aaoqHoM rrpellITOJIO)Ke·
HHH Herrpepb!BHOCTH TaHreHUHaJibHOro HarrpH)KeHHH 2). TeopcrHqe-
CKH JIOOOBOe conpOTHBJie1rne onpe,ueJIHeTCH no cpopMyJie
( 2μ +3μ')
D = 61tC'Vfl- 3μ. + 3μ.'
'.:::'. 4'ITC[J-'V, ecnu fl-'~ fl-,
(18)
fJI.e μ - B513KOCTb )KH,UKOCTH, a μ' -
BSl3KOCTb )KHJl.KOCTH IlOJl.HH-
Ma10merocH ny3biphKa.
Bee 3TO OTHOCHTCH K TeOpHH. Ha rrpaKTHKe )Ke Jl.JIH conpOTH-
BJieHHH, rrO-BH.ll.HMoMy, o6hil!Ho sepHa cpopMyJia (15), a He (18).
;:ho 3H8lJHT, Ol!eBH.ll .HO , '!TO rry3b!peK sener ce(fa TaK, KaK eCJIH Dbl
l) CM. [7J, n. 91 -9la, KJIK [17], rn. xr. § 1-3
2 ) Rybczynski \V. ,
Bull. Acad. Sci. Cracovie (1911). 40; Hada-
mard J. ,
Comptes Rendus, 152 (1911), l 7:l5. OrnornTeJILHO 9Kcnep1!MeHTaJib-
HhIX ir.attHhlX CM. B arr G., A manual of viscometry, Oxford, 1931, CTp. 190
H ir.aJihwe; Bryn T., Forschung Irzg., 4 (1933), 27-30; [II], T. 2, n. 74.

70
I',1 . II . llapaaoKCbl 8R3K020 rerteHUJI
OH 6&1JI TBepJJ.blM TeJIOM. TaKoe npornsope•me Me>K.11 .y TeopHeii H
3KcnepHMeHTOM MO>KeT 6b1Th H83B8HO napaJJ.OKCOM IlOJJ.HHM810IUe·
roc.H ny3blpbKa.
KaK npe.11.rroJJaraJJH Boa.11 . 1) H .npyrne aBTopbl, Ka*yIUa51CH TBep·
, llOCTh, B03MO>KHO, o6'bHCH.HeTCSI o6pa30B8HHeM TOHKOH (MOHOMOJJe·
KYJI.Hpttoii) nneHKH Ha nosepXHOCTH ny3hlpbKa H3 pa3JJHtIHhlX rrpH·
Meceii, H na nJJeHKa OK83hIBaeT conpOTHBJJeHHe .neq>opM8UHH 2).
0JJ.H8KO IlOJJH851 KapTHHa see eiue He HCH8.
Eiue 6onee :1q>q>eKfHhIM HBJIHeTcH cne.ny10ur.Hii napa.u.oKc.
IlapaJJ.oKc na.u.a101Uero ny3blpbKa. Ilpu aepTUtcaAb·
HOM epaaueHre reMneparypbt B rJ/Cua,cocru U3MeHeHUfl. noaepxHocr·
HOW HGTRrJ/CeHUR MOZYT npuaecru K TOMY, l{,TO ny3btpeK 6yaer ony·
CKGTbCf/., a He noaHUMQTbCR 3 ).
CTHrHB8HHe nosepXHOCTH ny3blpbKa no HanpasneHHIO K CTOpoHe
c 60JiblllHM nosepxHOCTHbIM HaT.H>KeHHeM 3aCT8BJJHeT nyablpeK B
BH3KOH >KHJI.KOCTH JJ .Blff8ThCSI B Hanpasnemrn y6hIB8HH51 noBepx·
HOCTHoro Harn>KeHH.H , T . e. B HanpasneHHH B03pacrnmrn TeMnepa-
TYPbJ. 3To SIBJJem,e Ka)KeTCH napaJJ.OKC8JlbHhIM TOJlbKO IlOTOMY, "<ITO
OHO TaK Heo6hltfHO, H IlOTOMY, tITO s MexaHHKe >KHJJ.KOCTeH llOtITlf
scer.u.a ycJIOBHO npHHHM8IOT nosepXHOCTHOe H8TH>KeHHe (KaK H
B513KOCTb) IlOCTO.HHHblM.
.
§ 33. «BTOpast• BH3KOCTb
KaK yKa3bIBanocb s § 19, npH 06blt1HOM BblBo.ne ypasHeHHH Ha-
Bbe - CTOKCa ( 1*) Mbl HMeeM .neJio c JJ.BYMH K03q><pHUHeHT8MH BSl3·
KOCTH A. H μ. Mo>KHO npHHSITb, "<ITO K03<pcf>HUHeHT B.H3KOCTH μ npH
cJJ.BHre H3MepHercsi JJ.JIH ret1eHHH IlyaaeiiJIH; rnrJJ.a ocraeTCH 3a.nat1a
H3MepHTb K03Q>Cf>HuHeHT A. H nposepHTb CJieJJ.CTBHH ypaBHeHHH ( l *)
JJ.JIH 3TOro K03q>cf>m.1 .HeHT8 "'·
KOTOpb!H, Bepo.HTIIO , 38BHCHT OT TeM·
nepaTyphI T H .u.aaneHHH p.
KaK 6bIJio cKaaauo B § 19, CToKc npocro npe.u.nonaraJI, tITO
"' = -2μ/3. 0 .1t.H8KO SICHO, tfTO HaJl.e)KHee BBeCTH B pacCMOTpeHHe Be·
JIHtIHHY μ' = A. + 2μ/3 H HCCJieJJ.OBaTb ee 3KCnepHMeHT8JlbHO. c (J>H·
3H"<leCKOH TOtIKH 3peHHSI A. H «BTopasi» B.H3KOCTb μ' He HMeJOT CMbl·
cna, noKa OHH He onpen.eneHbI ii He HaMepeHbI 3KcnepHMeHT8JlbHO.
I Bond W. N., Phil. Mag. , 4 0 9 2 7}, 889-898 . 0TJJH '1Hhlii o(faop eonpoca
cM. Mc Now n J. S., La Houil/e Blanche, 6 (1951). 701 -722 .
2) Conpornsnetttte MOlKeT 6b1Tb esnKHM HJJH ynpyrnM; cM. Cr id d I e D. W .
and Meader A. L., Jr. , / . Appl. Physics, 26 (1955), 838-842 H npHee.11eHHY10
TaM JIHTeraTypy.
3
) B ock M. J., Young N. 0., GoIdstein J. S., !. Fluid Mech.,
(j (1959), 350-356; CM. [17), CTp. 3)9.

§ 39. cBTopaa> BJl3KOCTb
71
A 3TO CJJ.eJJaTb He .nerKo. TaK, s norpaHHl!HbIX cnoHx c>KHMaeMou
lKHJJ.KOCTH 3Ha'lem1e Jt' He Hrpaer 60JlbllIOA po.nH, T8K KaK BeJIII·
l!HHa .necjJOpMaUHH C.D .BHra H8MHOro npeBbIWaeT Be.nHl!HHy C>K8THSI.
OT11acm no non npH'!HHe B reopen re11eenA c 6oJJbWHMH cKopoc-
THMH «BTOpOH» BSl3KOCTblO 06bll!HO npeHe6peralOT I) H eae6onee
Jl06pocoBeCTHble asropbl OCT8BJISIIOT OTKpblTblM Bonpoc O COOTHOWe-
HHH Ilyaccoea - CTOKCa μ' = 0.
9KcnepeMeHTaJJbHble onpe..a .e .nemrn μ' o6blllHO OCHOBbIBaJIHCb ua
H3MepeHHH 38TyxaHHSI 3ByKa, HO TeopeTHl!eCKOe HCTOJIKOBaHHe Ta·
KHX H3MepeHHH JJ.aJieKo He npocro. TaK, CToKc B csoetl" Teopun 3a-
TyxaHHS1 3ByKa He TO.nbKO npeJJ.nO.naraer, l!TO μ' = 0, HO Bee BHll·
M8HHe YJlMHeT TOJlbKO BeJIHl(l.fHe μ H, KpOMe roro, He YllHTblBaeT
pacceHHHe, Bbl3blBaeMoe TennonpoBOJlHOCTblO (TeDJIOBaSI JJ.Hcl><l>y-
3HH). IlocJieJJ.Hee 6blJIO YllTeHO I(Hpxro4>0M, KOTOpblH T8K>Ke Bbll!HC·
JIHJI (c 60JibWHM 3aBblWeHHeM) 38TYX8Hlle, ooyc.noBJieHHOe TpeHHeM
e norpaHHl!HOM CJioe ope pacnpocrpaHeHHH 3ByKa e rpy6ax 2),
npetieM YllHTb!Banac& TOJI&Ko BeJIHl!HHa μ. Ho, no-BHJlHMOMY, o6a
STH aeropa He paccMaTpHBaJIH 3aryxam1e 3ByKa KaK cpeJJ.CTBO
. ivrn
H3MepeHlrn eeJIHl!HHbl μ'.
XOTSI HHTepnperaUHSI 3Kcnep11Me1naJ1bHblX Jl8HHblX Bee eme HB-
JISleTCSI JJ.O HeKoropoA creneHH nporenopel!HBOH, c.ne .ny10mHe 4>aKTbl,
00-BHJJ .HMOMY, BDOJIHe pa3'bHCHeHbl • .U .nH HeKOTOpblX ra30B, TaKHX,
KaK He, Ar H N2, onbIT cornacyerrn c npe..l(no.no>KeHHeM ii' = 0 3).
Ho JJ.pyrne ra3bl, raKee, KaK 0 2 H C02, Aa!OT ropa3..a .o 60.nee pe3Koe
3aryxaeue 3ByKa s onpeJJ.e.neHHblX noAocax qacror 4 ).
B B03J1.yxe uenponopueoeaJibHO 6on&wee 34>cfieKTbI Moryr ObITb
Bbl3BaHbl He3Hal!HTeJibHOH OTHOCHTe.nbHOH BJia>KHOCTblO HJIH ueOOJlb·
WOH npHMeCblO C02, pasuo KaK I{ Dbl.nblO, a T8K>Ke wepoxoBaTOCTblO
CTeHOK ( B rpyoax). .UJIH OOJlbWHHCTBa )KHJlKOCTeH norJIOllleHHe
CHJlbHO 38BHCHT OT ll8CT0Tbl; KpoMe Toro, HeOOXOJJ.HMO rmare.nbHO
CJieJJ.HTb sa COJI.ep>KHMblM DY3blpbK8. TaK, npu OTHOCHTeJibHOM o(he-
Me nysblpbKa, paBHOM 0, 17 %' CKOpOCTb pacnpocrpaHeHHH 3ByKa
1) CM. [15), crp. 36-38, "(10], rJI. II. no noeo .!ly 3aMe'!aHnlt ornocareJT1,110
npe.11 .noJIO>KeHl!ll μ' = 0 CM. [7], n. 325, 328; [10), crp. 185, H lll] T. 1 CTp. 260
np11Me'!aHne.
·
•
•
•
2
) CM. [7], n. 359-360; l(irchhoff G., Pogg. Ann., 177(1868),177-193.
0TH~CHTeJtbHO tpeHHll a norpaHH'IHOM CJJOe CM. § 115.
) 3To MO>KHO CBll38Tb c .llOK83aTe.'lbCTBOM MaKCBeJIJla (nptt nOMOlllll MOJ!e•
KYJISlpHblX coot5palKeHHA, CM. § 34) Toro, '!TO .llJlll 11.11eaJibHOrO OAHOaTOMHOro ra3a
3). + 2μ = O; CM•• TaKlKe Ro c a rd Y., Hydrodynamique et theorle clnetiquc
des gaz, Affend1x I.
4
)Du A.W.,
Phys. Rev.,
6 (1898), 129-139 11 ti (1900) 65-74 ·
van Itterbeck A.,
Mariens P.,
Physica, 4 (1937), 207-215 ~
609-616; Markham J. J ., Beyer R. T ., Lindsay R. B ., Revs. Mod.
fllft.
1
~~
8
. (1951), 353-611; Marvin R. S . 11
.ap., J. Appl. Phys., 25 (195 4),

72
f A. //. llapaaoKCbl BJl.3K020 Te<teHUR
yMeHblllaeTCH B 5 pa3 I) H COOTBeTCTBeHHO yeeJJH4HBaercH 3aryxa-
HHe 3ByKa.
Jlerqe BCero o6'bHCHHTb 3aBHCHMOCTb 3aTyxaHHSI 3ByKa (YJJbTpa-
ssyKa) OT qaCTOTbl, BBOJJ.H BpeMJl peAaKCal{UU 2) (rHCTepe3HCHblff
9q>QJeKT) npH nepexoJJ.e MOJJeKyJJHpHOH 9HeprHH OT OJI.HOH co6cTBeH-
HOH qaCTOTbl K JJ.pyroH. 0JJ.H8KO pa3JJHqHhle aBTOpbl IlhlTaJJHCb co-
rnacosaTb .n.aHHbie Ha6mo.n.eHHH c COOTBeTCTBYIOlllHMH HHTepnpe-
TaU.HHMH «BTOpOH» BH3KOC1 H μ' 3), a cne .n.oeaTeJJhHO, H BeJJHqHHbl A
B ypaBHeHHH (l*) 4 ).
§ 34. MoneKym1puble scl>cl>eKTbl
MoneKyJJHpHoe crpoeHHe >KH.ll.KOCTeH o6'bHCHHeT MHOrHe HBJJe-
HHH B .ll.HHaMHKe peaJJhHblX >KH.ll.KOCTeH - He TOJJbKO HaJJHqHe oco-
6blX IlOJIOC qaCTOT npH nornomeHHH YJJbTpa3BYKOBbIX BOJJH, yKa3aH-
HOe s § 33. HacHJJbCTBeHHblM H HeecTeCTBeHHblM 6hIJIO 6bl HCCJJeJJ.O -
BaHHe MHOrHX TaKHX SIBJieHHH B paMK3X opTO,ll.OKCaJibHOH MeXaHHKH
KOHTHHyyMa.
HanpHMep, npH ou.eHKe TOJilllHHbl tppOHTa y.n.apHOH BOJIHbl Heo6·
XOJJ.HMO yqHThlBaTb speMH peJJaKcaU.HH B Tex }KH.ll .KOCTHX , B KOTO -
pb1X no.n.o6Hble MOJieKyJIHpHbie SIBJieHHH THila rHCTepe3HCa OKa-
3b!BalOT BJIHSIHHe Ha BeJIHqHHY «BTOpOH> BH3KOCTH ( CM. npHM. 2)
Ha 9TOH cTp.). (B KJiaccH4eCKOH TeopHH MexaHHKH KOHTHHYY·
Ma TOJllllHHa tppOHTa y.n.apHOH BOJ!Hb! npe.n .nonaraeTCSI pc!BHOH
HYJIIO.)
8 sepxHHX CJIOSIX 8TMOCq>epbl, r .n.e oqenb MaJia IlJIOTHOCTb (.na -
BJieHHe)' speMH peJiaKcau.HH npH M > 2 cymecTBeHHO BJIHHeT H8
BemfqHHY OTxoaa yaapHOt1. BOAHbt OT .ll .BH}KylllefOCS1 TeJia 6 ).
1) Wood A. B ., A textbook of sound, crp. 362 . no noaoJ.ly YJUIBIITeJib·
HhlX 9«p<j>eKTOB .11.aaneHHII CM. Lit o v It z T. A., Carne v a I e E. H., J. Appl.
Phys., 26 (1955), 816-820.
2 ) Kneser H. 0 ., Annalen der Physik, 11 (1931), 761-776; Eckart C.,
Revs. Mod. Phys., 20 (1948), 232 -235; Liebermann L. N ., Phys. Rev.,
75 (1949), 1415 -1422; Lighthill M. J. e paoore [20). pa3JI.. 4 (orHocHreJibHO
TOJilllHHbl ynapHoA BOJIHhl CM. pa3.II.. 6); Herz f e I d K. F ., L it o vi t z T. A,
Absorption and dispersion of ultrasonic waves, 1959.
1) Tis z a L., Phys. Rev., 81 (1942), 531-536; r y pea u" C. 6., 0,oKA.
AH CCCP, 55 (1947), 17 -19; Karim S., Rosen head L., Revs. Mod.
Phys., 24 (1952), 108-116; Trues de 11 C., I. Rat. Mech. Analysis, 4 (1953),
643-721.
4 ) no noeony H3noJKeHHblX a 9TOM naparpa«pe coo6pa}KettuA CM. C n e 3·
KHH H. A.,
.lluHaMHKa B113Koii Hec}KHMaeMoA )KHJI.KOCTH, M ., fTTH, 1955, rn . I
HJIaHJ.lay JI.
.ll..JIH«pwHuE. M.,
MexaHHKa cnnowHblX cpe.11. ,
H3Jl. 2 -e,
M., fTTH, 1954, rn. II H § 78 e rn. VIII. -llpuM. nepe8.
5) CM.,
HanpHlllep, Schwartz R. N ., Eckerman J., I. Appl. Phys.,
27 (1956), 169-174. EcnH (51,1 9TO 6blJIO pe3yJibTaTOM MrHOBeHHOH AHCCOUHaU.HH,
TO BeJIH'IHHa 3TOrO paccTOIIHHII He 3aBHCeJia (51,1 OT DJIOTHOCTH (§ 73).

§ 34. MoJte,cy.1111pHble 3i/Jc/JeKTbt
73
KpoMe TOro, MOneKynsipH8Sl D:HCCOUH8UHSI, peKOM6HH8UHSI MO-
JieKyJI H HOHH3aUHSI BJIHSllOT Ha TOJIIUHHY y.napHOH BOJIHbl B rnnep-
3BYKOBOM noroKe 1); .n:eHCTBHTeJibHO, OHH B 3HaqnreJibHOH Mepe
BJIHSIIOT Ha JlBH:>KeHHe >KH.llKOCTH s CJiytJae, Kor.na npH o6bltJHblX
arMocipepHbIX ycJIOBHSlX tJHCJio Maxa M > IO. TaK, so3.n:yx cp.n:ep-
mHT 1% NO npH 2000° K H 10% NO npH 3000° K. Dpn reMnepary-
pax cBhIUJe 11 000° K craHOBHTCSl omyrHMoii noHH3aUHS1.
no 9TOH npHllH:te npn peaJIHCTHlJeCKOM no.n:xo .ne K H3ytJeHHIO
rnnep3BYKOBhIX y.napHb!X sonH see s 60JibUJeii Mepe Ha.110 ytJHTb!·
B8Tb XHMHlJeCKYIO ipH3HKY 2).
Opro.noKcaJibHasi MexaHHKa KOHTHHYYMa He B CHJiax raK>Ke
o6"hHCHHTb ipH3HlJeCKOe HBJieHHe aut/)t/Jy3UU H o6paTHOe eMy- pa3-
aeAe1tue COCTaBHOH >KH)lKOCTH HJIH raaa Ha CBOH KOMilOHeHTbl, Ha-
npHMep OT.lleJieHHe CJIHBOK OT MOJIOKa. 8 H.lleaJibHOM KOHTHHyyMe
or.neJieHHe H30TOilOB He 6bIJIO 6bl B03MO>KHO HH c IlOMOlllblO raao-
BOH .n:mpipyann, HH s ueHrpnipyrax, HH B conn ax 3).
CKoJIb>KeHH.e
.lK H .n Koc r n. CosceM HHatJe ncnoJihayrorcsi
MOJieKyJIHpHble npe.n:craBJieHHSI npH o6"bHCHeHHH HapyrneHHSI rpa-
HHlJHOro ycJIOBHSl (npHJIHnaHHSl >KHJlKOCTH) (6)' Kor.n:a cpe.n.HSlSl
D:JIHHa cno6o.nHoro npo6era MOJieKyJihl cpaBHHMa c MaKpocKonHtJe-
CKHMH pa3MepaMH. Mo:>KHO OTMeTHTb npH 9TOM rpH Ba:>KHb!X qacT-
HbIX cJiyqasi: · retJeHHe qepea UJeJIH, cBo6o.nHoe na.neHHe MeJibtJaii-
lllHX KaneJib (onb!T MHJIJIHKeHa C K8nJieii M8CJia) H ropMo:>KeHHe
cnyTHHKa. Bo Bcex 3THX CJiytJaSlX BeCbMa 3aMeTHO OTKJIOHeHHe OT
3aKOHOB MeXaHHKH KOHTHHYYM8 4): H86JIJo.n:aeMble ycHJiliSl C.ll .BHra
3H8lJHTeJibHO MeHbllle, 'leM npe.n:cK83blB8eMble ipopMyJiaMH ( 13)
H (15).
Jierqe Bcero HCTOJIKOBaTb YK838HHhle SlBJieHHSl K8K Bbl3B8HHhle
tJaCTH'IHb!M HapyrneHHeM YCJIOBHSI (6). B npe.n:eJibHOM cJiyqae «aep-
KaJibHoro orpa:>KeHHSl» scex MoJieKyJI (yroJI na.neHHSl paBeH yrny
OTpa.>KeHHH) Hanpsi>KeHHe C.l{BHra Ha rpaHHUe, OtJeBH.ll.HO ,
.ll .OJI:>KHO
paBHHTbCH HYJIIO. TaKHM o6pa30M, ycJIOBHe (6) JIOnllJHO 38MeHHTb
ycJIOBHeM HenpepbIBHOCTli HOpM8JlbHOH COCT8BJIHIOIUeii CKOpOCTH
(rJI. I, (7)), CBOHCTBeHHblM HeBH3KOMy TetJeHHIO. TaK K8K
1
) CM. Light hi 11 M. J., J. Fluid Mech., 2 (1957), 1-32, a TBK>Ke [14].
2
) CM. Nature, 178 (1956), 343-345; De a I W. E., J. Appl. Phys., 28
(1957),. 782. U.eHHblH o(53op HCCJJe.U.OB8HHH, npose.u .eHHblX s Te<1euue BTOpOH
Mnpo~ou BOHHbl, c.u .e .1a11u Pen n e y W. G .,
Pike H. H. M. Progress in
Physics, 13 (1950), 46-82 .
'
3) no noeo.u.y a3p01lHH8MH'lecKHX MeTO.IJ.OB OT.ZleJJeHHII H30TOROB (c nOMOIUblO
conna) CM. Leroy M., Nucleonics, April, 1960, crp. 68.
4
) K n u d sen M., Annalen der Physik, 34 ( 1911), 593-656. O ue.u.asuux
0~3opax c 6u6nuorpa<j>HHMH cM. Patterson G. N .• Molecular flow of gases,
Wiley, 1956, rn.V; a TaK>Ke Schaaf S., rn. 9-10 B Heat transfer Univ
of Michi1ia1; Press, 1953; [16], rn. 8.
'
'

r .A , II . llapadotcCbl. BJl31COZO Te'leHUR
v2 ui = iJ ( V2U) /oxi = O JlJHI J1106oro 6e3snxpesoro ttec>KHMaeMoro
lIOTOKa, JII06oe peweHHe 38Jl8l!H HeHMaHa (Tel!eHne )KyKOBCKOrO
HJIH 3irnepa) .ll.OJI>KHO YJlOBJieTBOPHTb ypaBHeHHHM Hasbe - CroKca
(5). TaKHM o6pa3oM, yqer sepKaJibHoro orpa>KeHHH .n .omtceH 6hlJ1
6bl Bbl388Tb DOJIHblH nepecMOTp TeopHH BH3KHX TelleHHit
nJISI o6nSICHeHHSI 9KcnepnMeHT8JlbHblX ¢aKTOB BMeCTO DOHSITHSI
sepK8JlbHOro OTpa>KeHHSI BBOJlSIT KB83H9MDHPHlleCKHH «K09¢¢uuneHT
8KKOMO.ll.8U:HH>. IloJib3YSICb T8KHM K09<p<pHU:HeHTOM H YllHTbl88SI no-
cry.linpyeMOe B KHHeTHl!eCKOH TeOpHH ra30B KOHel!HOe 3H8l!eHHe
cpe)lHeH JlJIHHbl CB060JlHOro npo6era, t5blJIH DOJIYlleHbl pe3yJibT8Tbl,
B o6meM cornacy10ll.lHecS1 c )laHHhIMH 9KcnepnMeHra.
«CKOJlb>KeHHe> B Te11eHHH pa3pe>KeHHOro ra3a HeJib3SI cMeWH-
B3Tb c npe)lCTaBJieHHSIMH )leBSITHa)luaToro CTOJieTHSI 06 06IUeM
CKOJib>KeHHH Ha rpaHHUe BeCbMa rJla)lKHX TBep.ll .bIX TeJI (HanpHMep,
Hg no cTeKJ1Y). TaK, CroKc 1) CllHTaJI, llTO «CKOJib>KeHHe> JlOJl>KHO
HaCTYD8Tb Hal!HHaSI C onpeJleJieHHOH CKOpOCTH, TOrJla KaK MHOrHe
JlpyrHe BblJlaIOll.lHeCSI y11eHble 803Jlep>KHB8JIHCb OT BblCKa3bl88HHH
no 9TOMY sonpocy. Bsu.a:y MHornx oco6eHHOcTeii <l>H3HKH nosepx-
Hocreii TaKOe DOJIO>KeHHe He CJIHWKOM YJlHBHTeJibHO. no)lTBep>KJle-
HHeM B3rJIHJlOB CroKca Morna 6b1Tb H npeJ1.nonaraeMaS1 aHa.11orHS1
c TpeHHeM TBepJlbIX TM, npH KOTOpOM HanpH>KeHHe C.ll .BHra 't orpa-
HHl!eHo npOH3BeJleHHeM DOCTOSIHHOH μ < } Ha HOpMaJibHOe Jl8BJie-
HHe. na>Ke B HaCTO.Hll.lee BpeM.H, HeCMOTp.H Ha TO 11TO DOJlaBJI.HIOll.lee
l!llCJIO <paKTOB CBHJleTeJibCTByeT npoTHB aHaJIOrHH c DOHSITHeM 06-
mero CKOJlb:>KeHHSI 2). sceo6meii H a6COJIIOTHOH ysepeHHOCTH B 9TOM
sonpoce noKa ne JlOCTHrnyro.
§ 35. 81,HOJlbl
KpaTKO H3JIO>KeHHble Bb!We <paKTbl .SIBJISIIOTCSI cepbe3HblM .!l.080 -
JlOM B non1,3y npHrO.ll .HOCTH ypasHeHHii Hasbe - CToKca JlJIH noro-
KOB HeaHCUMaeMblX B.H3KHX >KHJlKOCTeii, K KOTOpb!M OTHOCSITCSI Te11e-
HHSI o6bll!HblX ra30B ff >K'HJlKOCTeH npH CKOpOCTSIX, 3Ha11HTeJibHO
MeHblIIHX CKOpOCTH pacnpocTpaHeHH.H 3ByKa (T. e. eCJIH M < 0,2).
0.ll.HaKo JI.JI.ff 60JibWHHCTBa npHJIO>KeHHH HeJib3SI DOJiaraTbC.H Ha
npas.L1 .ono.L1 .06Hble rnnore3bl, nepe11ncneHHb1e B § I, xorn 9TH rHno-
Te3bl B Jlpyrux YCJIOBHSIX MoryT OK833TbCSI none3HblMH. Ilo9TOMY
oco6eHHO npn paccMOTpeHHH Typ6yAeHTHOCTU Tpe6yeTCSI BeCbMa
1) [13], T. I, cTp. 96-99. CM. TaM )l(e H CTp. 186.
2) CHCTeM3TH3HpOB3HHblii o(S3op 4>aKTOB OTHOCHTeJJbHO o6mero CKOJJb)l(eHHll
CM. B pa6oTe [3], T. 2, cTp. 356-361. OoCJJeJlHHe JlSHHble CM. Brock man M. It,
Nat. Bu. Standards Rep., 4673 (1956): Sch n e 11 E.. J . Appl. Phys., 'D (1956),
1149-1152; Debye P. and Cleland R L. , r aM ,1<e, 30 (1959), 843-849 .

§ 35. Bb10DrJ1>1
75
8KKypaTHOe np1tMeHem1e CT8THCTH'leCKOro aHaJHl3a Ha Bb!COKOM
MareMaTH'- leCKOM ypoBHC.
TaK, BMecTo HcrroJib30BaHHH yp aBHeHHH HaBbe - CToKca rrpH
cpopMyJrnpoBKe O'- !eBHJlHb!X KpaeBb!X 38.l(aq (HarrpHMep, 38 ,Ua\lH CTa-
l~HOHapHoro Te'- leHHH) HY:>KHO o6p amaTbCH Herrocpe.nCTBCHHO K (jJu-
3U~eCKOU 8eUCTBUTeAbHOCTU JlJIH ITOCT8HOBKH COOTBeTCTBYIOIUHX
KpaeBbIX 38JI,a 'l.
noJI,o6Hoe IlOJIO:>KeHlle HHKOHM o6pa30M He HBJIHeTCH qeM-TO HO-
Bb!M B aHaJIH3e, XOTH 06 3TOM qacro 3a6brna10r. TaK, OTKpbIBaH
DepBbIH Me:>K.n:yHapo,nHbIH MaTeMaTHtteCKHfi KOHrpecc, DyaHKape
CKa3an: «KaKHM 6b1 6oraTbIM HH 6blJIO Boo6pa:>Kem1e qeJioBeKa,
npHpo.na eme 6oraqe B TbICHtJY pa3», H .nanee: «Ka:>K.n:aH cpH3Htte-
cKaH Teopm1... no-HOBOMY OCBemaeT ypaBHeHHH (B ' -!aCTHb!X npOH3-
BOJlHb!X) ... 6e3 3THX TeopHH Mb! He 3H8JIH 6hl, ttTO TaKOe ypaBHe-
HHH B ' -!8 C THb!X npOH3BOJlHb!X» 1). C Tex nop 3TH OCHOBHble HCTHHbl
no.nrnep>KJl8JIHCb H .npyrnMH Bb!Jl8IOI.UHMHCSI MaTeMaTHKaMH, H HX
BcerJI,a CJieJI,yeT IlOMHHTb.
MeHee HCHO, KaKHe CBOHCTBa MO>KHO 38KOHHblM o6pa30M npH-
DHCaTb C;JfCU.MaeMblM BSI3KHM >KHJI.KOCTSIM. XoTH MO>KHO HaJI.eHTbCH,
'I TO
KaKoe-TO BHJl.OH3MeHeHHe ypaBHeHHH Haebe - CTOKCa 6y.neT
xoporno COOTBeTCTBOBaTb qJH3H'- !eCKHM cpaKTaM, OCTaeTCH HeHCHb!M,
KaK YBH3aTh 3TO BHJI.OH3MeHeHHe C repMOJlHHaMHKOH.
llJIH IlOJIHOH CTporocrn HY:>KHO yttHTbIBaTb KaK TennorrpoBo.n -
HOCTb H H3JIY' -!eHHe, TaK H HarpeB 3a C'- !eT rpeHHH H H3MeHeHHe BSI3-
KOCnl H IlJIOTHOCTH C H3MeHeHHeM TeMneparyphl. CTporoe pernem'. !e
B03HHKalOI.UHX rrpH 3TOJ\1 KpaeBblX 3a.l(a' - !
11 JlOKa3aTeJibCTBO Toro,
' -!T O 3a.l(a'IH KoppeKTHO IIOCTaBJieHhl, IlO-BHJlHMOMy, [lQl[Tl{ 6e3Ha,Ue)i{-
HaSI rrpo6JieMa. CTOJih >Ke, ecJIH He 6onee, TPYJlHbIM JI.eJioM rrpe.n:-
CTaBJIHerrn CTporoe rrpHMeHeHHe MeTOJlOB reopHH B03Mymemrn, 060 -
CHOBbrna10mee npeHe6pe:>KeH11e or.n:eJihHhIMH nepeMeHHbIMH. .[J.o-
CTH)i{eHHe ycrrexa B 3TOM CJly' - !ae 6yiteT 38BHCeTb OT HCITOJib30BaHHSI
rHilOTe3 (A)-(F) 113 § 1 H .n:pyrnx nono6Hh!X 3BpHCTH'leCKHX npe.n: -
[IOJI o:>KeH HH.
0.n:HaKO 11.0BOJ!bHO o6o6meHHH! Tenepb, He CBSI3b!B8SI ce6si crpo-
rofi rrporpaMl\!011, Mb! rrepe11.n:eM K H3y4eHHIO Tpex pa3JIH'lHb!X KOH-
uemi:1111 rnitpon1rnamrKH, l!T06b1 nonytJHTb 6onee rJiy6oKoe npe11. -
craBJie1me o TOM, Ka1rnro po.na coo6pa>KeHHH xapaKTepirny10T cy-
mecrno Bonpoca. Mb1 HMeeM B BHJly reopmo CTpyft, no.no6J.1e H
npHcoe.n:HHeHHbie M acch1.
1) Poi n care H., Proc. First Int. Math Congress, Zurich, 1897, CTp. 81 -
90. noArnepJKJieHHR 3TOii T04KH JpeH11R CM. Ha clam a rd J. ,
Lectures ou
Cauchy's prcblem, Yale Uni\·.
Press, 1923, crp 2311CourantR.,
Proc.
Ele\'enth Int. M;i!h. Congress, Cambridge (USA), 19:ill, T . 2 , cTp. 278 .

fJIaeaIII
CTPY'H, CJI E.l].bl H KAB HTAU.HSI
§ 36. Pa3pb18Hble Te'leHHH
IlpH .11 .BH}KeHHH peaJibHblX }KHJlKOCTeH c MaJIOH B513KOCTb!O o6hl4·
HO MO}KHO 3aMeTHTb, llTO IlOTOK CTpeMHTCH OTOe/lUTbCJl OT TBep.nbIX
CTeHOK, oco6eHHO y OCTpb!X yrnoB. 3ro 6bIJIO y}Ke OTMetJeHO a § 8,
r.ne Ha pHC. 2, B H306pa}KeHO TaKoe TetJeHHe, a TaK}Ke B § 29.
llmtORUJQJICff :J1CUiJ1<ocmb
-.-v
---
-- Cm.J!.i .H -
f/OffOffU[QffCH :HCUUlfOClll/:J
a
==========
TuepiJbtt1
qunu1tilp
i llOlfORIJlUUCJI
cnia
----------
-- --- --- --
p H c. 9. a - npRM311 crpy11 H.neaJJbHOH lKll.llKOCTH; 6 - CJJe.ll 0033,Ali nony-
~H JlliH.llP3 B H.neaJJbHOH lKH.llKOCTH.
MaTeMaTHtieCKHe MO)J.eJIH no.no6HbIX TetieHm1 c OTpblBOM MO}KHO
ll.OBOJlbHO JierKO nocTpOHTb, HCil0Jlb3YH ypaBHeHHH JlBH}KeHHH 3ii-
Jiepa .11.JIH HeBR31COU }KH.11 .KOCTH. OcHOBHaH u.neH COCTOHT B TOM, l!TO
.ir.onycKaeTCH CKatIK006pa3HOe H3MeHeHHe CKOpOCTH npH nepexo.ne
qepea JIHHHIO TOKa, tJTO 51BJISleTC51 rpy6b!M HapyrneHHeM rHilOTe3bl
(E) H3 § 1. npoCTble npHMepbl TaKHX Tel!eHHH cxeMaTHtieCKH H30-
6pa}KeHbl Ha pHC. 9 . B 3THX TetJeHHHX ace JIHHHH TOKa napaJI-
JieJibHbl .npyr .npyry, a o6Jiacrn paaHoMepHoro TetJeHHH OT.!leJieHbI
OT o6JiaCTeii CTOHlleH BO.II .bl /lUHUJl.MU TOICa, npH 11epexo.ne qepe1
KOTOpbie CKOpOCTb H3MeHHeTCSl CKll'iKO.M . Ha pHC. 9, a H306pa}KeHa
H.neaJIH3HpoaaHHaH 6ecKoHetJHaH crpyR; nocTyna10IJJ.aH B o6JiaCTb
Heno.nBH}KHOH BO.II.bl H3 rpy6bl npOH3BOJibHOrO nonepetJHOro cetJeHHH,
a Ha pHC. 9, 6 H306pa}KeH paBHOMepHblH IlOTOK, OTpbIBaJOIJJ.HHCH OT
noJiyLI.HJIHH .!lpa co crnpoHbI cpe3a H o6TeKaJOIJJ.HH 3acTOHHbIH cAeiJ
noaa.nH 3T0f0 noJiyU.HJIHHJJ.pa. B o60HX cnyt1aHx .naBJieHHe MO)KHO
CtJHTaTb rn.ir.pocTaTHtJeCKHM.
no onpe.neneHHIO, B 11 .n .eaJJbHOH HeB51 )!<OH }KH.11 .KOCTH ycHJIHe
C.11 .BHra paBHO HYJJIO; cne.noBaTeJibHO, Heo6xo.nHMOe 11 .!lOCTaTOtJHOe

§ 36. Paspbl8Hb1e re'leHuR
77
---- ----
ycJJODlle paBHOBeCHH Ha JIHHHH TOKa, HBJIHIOllleHCH JIHIIIICH pa3pbl-
ea. -
9To HenpepbtBHOCTb aasAeHUfl npu nepexoJJ.e 4epe: Hee.
EcJIH mm1rn TOKa orpam14ueaer u11eamBupoaaHHblH CJie.!l u.1rn
KaKy10-m160 JJ.pyrylO o6JiaCTb, 3anoJ1HeHHYIO HenOJJ.BH>KHOH >KHJJ.KO-
CThlO ( «MepTBaH Bona»)' a CHJJa TSUK€CTH Y4T€Ha cornacHO TeopeMe 1
H3 § 21, TO ycJIOBHe ttenpepbIBHOCTH 11aBJ1€HHH paBHOCHJlbHO ycJJO-
BHIO noCTOHHCTBa Jl8BJl€HHH B paccMaTpHsaeMOH o6nacrn. no3TOMY
BBHJlY HenpepblBHOCTH .naoJJeHHH Ha JIHHHHX TOl<a, orpaHHl!HBalO-
IUHX CJleJJ.. JlaBJJeHHe JlOJDKHO 6b1Tb nOCTOHHHblM. JlHHIHI TOKa, Ha
KOTOpblX CKopoCTb H3Mem1ercH CKal!KOM, a .nasJJeHHe nOCTOHHHO, Ha-
3bIBaIOTC51 CB06oaHbL.Mll AUHUJlMU TOKa.
I
I
t
I
-,-.
a.
s
PaJaenJ110ll/OflC}f O
00 --~- -.!;.JI
llUHl/51 /lltJ!fQ
s'
Cnea
P II c. 10. a - KpyrnaR ctpyR; 6 - cneJJ. D03MH
.llHCKa.
.
-
B CHJJY ypaBHeHHH 6epHyJIJIH (8*) rn. I, eCJIH Bee eme npette-
6peraTb CHJIOH Tff)KeCTH, CKOpOCTb ocraercH nOCTOHHHOH BAOJlb JJIO·
6ou CB060JJ.HOH JIHHHH TOKa npH craunoHapHOM Tel!eHHH, H Hao6o-
por: v dv = -dp/p = 0. 3TO JJ.aer l!HCTo KHHeMaru11ecKoe Kpaeeoe
ycJJOBHe JJ.JJH craunoHapHblX rel!eHHH, orpaHHl!eHHblX cso6oJJ.HbIMH
JIHHHSIMH. BMecre c lf>opMyJJaMH § 5 OHO onpeJJ.eJIHeT cne.11y101UYIO
KpaesyIO 3a,naqy Teopuu noreHUHaJJa.
3a.D.a4a re.11bMroJIbua. ilJJH aa.11aHHoro npenHTCTBHH R
H3HTH DOTeHUHaJJ CKOpocreu, YJJ.OBJJeTBOpHIOIUHH ] ) ypasHeHHIO
V2U = 0 BHe npemlTCTBHH R H BHe o6JJaCTH «MepTBOH 80.Llbl» R.;
2) ycnosnIO au;an = O Ha rpaHuuax npenHrcrsnH R u o6nacru R1
H 3) ycnoBHIO IVU 12 = const Ha rpaHnue o6JJacru R 1.
3aMeTHM, l!TO nocne.11Hee Kpaeeoe ycnooue He.1twteu1to. 3aMeTHM
TaK>Ke, 'I TO TO D O JIOrHH Tel!eHHH OCTanacb Heonpe)].eJieHHOH; Ha npaK-
THKe ee 3a.na10T HCXOJJ.SI H3 HHTYHTHBHblX npe.ncrasneHHH HJIH 3Kcne-
pnMeHTaJibHbIX .11aHHblX (rnnoTe3a (D) H3 § I). nee T8KHe TOOOJJO-
rHH Te'leHHH cxeMaTH'lecKH H306pa>Kelibl Ha puc. 10 . Ha 3THX
pucyHKax noKa3aHbI «crpyH», BbITeKaIOIUaH H3 Kpyrnoro oTBepcTHH
B nnOCKOH CTeHe, H «cJie.ll» 3a JJ.HCKOM.
TetJeHHH, YJlOBnersopHIOIUHe yKa3aHHblM YCJIOBHffM 1)-3). T . e.
peweHHH 3aAatIH renbMronbua, e nocne.ny10meM Mbl 6yJJ.eM Ha3bl·
BaTb TetteHtulMU f eAbMZOAb({a.

78
r A. ///. Crpyu, c. .teiJbl u ,casura14uR
B .neikTBHTeJlbHOCTlf >Ke HHKTO eu.1.e He cyMeJJ .narb TO'IHYIO Ma·
TeMaTHll.eCKYIO rpaKTOBKY yKa3aHHblX Bblllle JJ.BYX re,1eHHll feJibM·
rOJlhUa, CM , § 49. 0.ll .HaKO aeaJIOrHll.Hble reqeHttSI JJ.JISI nJJOCKOro
CJiyll.aSI, T . e. crpyH, BbJTeKBIOlll8SI H3 meJIH, H cJJe.n Il038.ll .H IlJIOCKOH
DJiaCTHHKH, MO:>KHO noCTpOHTb .11.0BOJlbHO JierKO. TeopHSI 3THX CTJIO·
CKHX Tell.eHHH feJibMrOJibUa 6y.D.eT npe.D .MeTOM HCCJie}lOB8HHSI B
§ 37-39 1).
§ 37. fo.l{orpacln.r e BHJJ.e nonyKpyra
B o6meM CJJyqae JIOKaJJbHO 6e3BHXpeBble HeC>KHMaeMbie DJIO·
CKHe Tell.eHHSI xapaKrepH3YIOTCSI cymecTBOBaHHeM KO.M.nAeKCHblX no-
reHt4UaA08 W = U + iV. 3 .necb U - noreHuHaJJ cKopocreA, a V -
l}>yHKUHSI TOKa. l(OMDJJeKCHblH noTeHIJ.H8Jl W CCTb aHaJIHTHll.CCKaSI
l}>yHKUHSI KOMnJJeKCHOH nepeMeHHOH Z = X + iy, xapaKTepH3YIOIUeii
DOJIO>KCHHe TOll.KH, a ee npoH3BOJJ.H8SI
dW ,.
•
dz =-.=u- tv
(1)
npe.l{CT8BJISICT co6ou conpR,iceHHOe 3Ha~eHue KOMDJICKCHOH CKOpO·
2)
•
'"*
dx
dy
CTR
u+iv= .,, ' r.u.e u=tit.
'() = dt.
EcJIH H3BecTeH DOTeHUHaJJ W = f (~) KaK KOMnJieKCH8SI aHaJIH·
THll.ecKaH l}>yHKUHSI ~.
TO, CJJeJJ.OBaTeJJbHO, MO>KHO onpe.u.eJJHTb z B
BHJJ.e aHaJIHTHll.CCKOH l}>yHKUHH OT ~.
T. e.
(2)
H CJJeJJ.OBaTCJlbHO, MO>KHO (B npHHUHDe). HCKJ1!01IHB ~.
HaHTH w(z).
MTaK, JJ.JJH onpe.u.eneHHH cTauHoHapHoro DJJocKoro relleHHH feJJbM-
roJibUa )l0CT8T01IHO 3H8Tb q>YHKUHOH8JlbHOe COOTHOilleHHe W = f (~).
I<aKOB BHJJ. 3TOH <l>YHKU.HH B cJiyqae DJIOCKHX TC4CHHH, npnse.n.eHHblX
Ha pHc. 10, MOJKHO .nora .u.arhcH no ro.norpaq>aM paccMatpHsaeMblX
reqeuui%.
f ooozpa</JoM nnocKoro reqemrn Ha3hIBaercH reoMerpttt1ecKoe Me·
CTO TeX 3Hall.CHHH
~.
KOTOpbie .11.eHCTBHTeJibHO )lOCTHraJOTCSI B 3TOM
Tell.eHHH. Ha pHc. 10 nerKo Bl.f .lI.eTb, 1ITO ro)lorpact>bl COOTBeTCTByJO-
lUHX DJIOCKHX TelleHHH (ecJJH OHH cymecrsy10T) .11 .0Jl>KHbl 6b1Tb
noAyKpyzaMu. 9ro cneJJ.yer H3 roro, llTO arg ~ (HanpasneHHe reqe-
HHH) - BeJIHl!HHa, DOCTOSIHH8SI Ha DJIOCKHX nJiaCTHHK8X ( q>HKCHpo-
1) 8 pyccKoA n11TepaType npKHHT TepMHH creopK~ crpyb, B 9TY TeopHIO
BXO)lHT H3}"1eHHe ecex re11eHHH co ceo6o)lHblMH noeepxHOCT51Mll, Ha KOTOpblx
J18BneRHe nOCTOIIHHO. -llpuM. peiJ.
2) 8 pycCKoA JIHTeparype KOMnneKCHOA CKOpOCTblO H83b1Baerc11 caMa BenH-
'IHHa dw/dz=u- iv. -
llpu1,1, pea.

§ 37. foooipacpbl 8 BUOe noAy,cpyea
79
B8HHblX rpamiuax), B TO apeMSI KaK ,~ ,
( CKOpOCTb TetJemrn) IlOCTO-
S!HHa B)lOJlb CB060)lHblX rpam:1u, KaK IlOK838HO B § 36.
C JJ.pyrofi CTOpOHbJ, o6JiaCTb w. HJIH reoMeTpH'leCKOe MeCTO
3Ha'leHHH, npHHHMaeMbIX B JJ.8HHOM Te'leHHH BeJIHlJHHOH W, orpaHH-
qeHa JIHHHSIMH V = const (mrn1m TOKa), napaJIJieJihHblMH JJ.eHCTBH-
TeJibHOH OCH u. Ha pee. 10, a - 3TO 6eCKOHe'IH8SI JlOJioca . .[I.JISI cJiy-
qaSI Ha pee. 10, 6 - 3TO nonynJIOCKOCTb, pa3pe3aHH8SI B)lOJlb IlOJIO-
)l{HTeJibHOH OCH U (ecJIH BbI6paTb IlOCTOSIHHYIO HHTerpupoBaHHSI a
w= f cdz TaK, 'IT06bI s KpHTH'leCKOH TO'IKe 6bIJIO w = 0).
AJlIlapaT KOHIpOpMHOrO OTO6pa»<eHHSI. nycTb Te-
nepb <I> - n106oe Tel!emie, HMeIOlll.ee ro .11.orpacpoM nonyKpyr ( cJie)lo-
aaTeJibHO, orpaHH'leHHOe CB060JI.HbIMH JIHHHSIMH TOKa H npS!MOJIH·
HeRHbIMH CTeI-IK8MH). MbI MO)KeM TaK Bbi6paTb OCH KOOP.llHHaT, 'ITO
aeJIHlJHHa s 6y)leT npttHHMaTb .11 .eiicTBHTeJibHbie 3HatJeHHSI Ha Heno-
)lBH>KHOH rpa1rnue, H TaK BbI6paTb eJI.HHHUbl H3MepeHHH, l!TQ Ha CB0-
60.ll .HOH rpaHttue 6yJI.eT ICI = 1. 3aTeM c noMOlll.bIO npeo6pa3oaaHHH
a = (C + s- 1) /2 OT06pa3HM o6JiaCTb rOJI.Orpacpa Ha HH>KHIOIO noJiy-
JlJIOCKOCTb Im {a} < 0. Km1cpopMHOe npeo6pa30BaHHe HaH6onee 06-
lll.ero B~JI.a, oTo6pa>KaIOlll.ee o6JiaCTb ronorpacpa Ha HH>KHIOIO nony-
nnocKOCTb, sa .11.aeTCH cpopMyJJoii
aa+b
a(~2 +1)+2bC
T= ca+d = c(C 2 +1)-f.2dC • ad> be,
(3)
r.11 .e a, b, c, d - neikTsHTeJihHbie 'lHCJia.
C JI.pyroi% crnpOHbI, o6nacTb W mo6oro O.llHOCBH3Horo Te'- le-
HHH, orpaHH'leHHOl'O JIHHHHMH TOKa, eCTb 0606meH1IbIH «MHoroyrOJib·
HHK», O)lHa HJIH 60. 'lbIIIee 'IHCJIO aeprnttH KOTOporo paCJlOJIO)KeHbl Ha
6eCKOHe'IHOCTH H ace CTOpOHbl KOTOporo napaJIJieJibHbl )leHCTBH-
TeJibHOH ocn. CJie)loBaTeJibHo, MO)KHO oTo6pa3HTb o6nacTb T, onpe-
JJ.eJisi:eMyIO COOTHOIIIeHneM (3)' Ha o6JI8CTb w npH JlOMOlll.H JlO)lXO-
JJ.Sllll.ero (KOHq>OpMHOro) npeo6pa30B8HHH lllBapua - KpHCTO<pcpeJISI:
err (T-BJ)
dl!_
-
R(T) -
1
(4)
dT-
-
II(T-Tk) •
k
rJJ.eCHBj,T,. -
iJet1cTBUTeAbflbte napaMeTpbl ([4], CTp. 370). Mbl
BH)lHM, 'ITO )lJISI JIJ06oro O)lHOCBSl3HOro re11eHHS! <I>, y KOTOporo 06-
JiaCTb roJJ.orpacpa ecTb JlOJiyKpyr, MO)KHO 38JlHCaTb W B BH)le
f A (C) dC, HCilOJib3YSI <l>OPMYJibl (3) H ( 4), r)le A (s) - pal{UOHU/lb-
H(I.Jl <pyHKl{UJl.
PacCl\rnrpeHHbifi BbIWe MeTOJJ. M02KHO nerno o6o6lll.HTh Ha
cny11afi, KOr.D.a o6nacrb ro,D .orpacpa eCTb KpyroBOH ceKrop c yrnoM,

80
[A. Ill. Crpyu, cM(}i,i u 1U1Bura14U.11
npu eeplllHHe paBHhlM Tt/n ([17)), rn. II). B 3TOM cny•-iae npeo6pa3o-
eaHHe ~n oro6pa>1caeT o6nacrb ro.11orpatpa Ha nonyKpyr H, cne.110-
eaTeJibHO, COOTHOllleHHe
a (C111 +1)+2bCn
T= c(c1n+1)+2den
npe.11craensier co6ou oro6pa>KeHHe o6nacru ro.11orpact>a Ha nonynno-
CKOCTb. A
.1 1aJ1bII1e Mbl .11eiicreyeM, KaK B npe.D .bl.llYllleM cnyqae.
§ 38. HcTeqeuae ctpya H3 m.ena
renbMroJibU. [27] nepBblH npHMeHHJI e 1868 r. onHC8HHblH Bblwe
annapaT K cny11aio nnocKoii crpyu, BblTeKaiomeii H3 menu, cM.
puc. 11, a. B 3TOM cny,rne, Bb16pae e.D .HHHU .Y
.n nHHbl TaK, 11To6hl cKa-
110K V npH nepexo.11e 11epe3 oreepcTHe 6bln paeeH Tt, Mbl MO>KeM B
ctiopMynax § 37 noJio>KHTb W = In T, T = ew. dW/dT = l/T.
0
00
s'
0
no
00
s
a
B
Pac. 11. nnocKu crpya, e1,1reKaroll{aa H3 menH .
.lI.n .H Bbl6opa BeJJHIIHH a, b, c, d paCCMOTpHM 3aBHCHMOCTb Me>KJJ.Y
t H w B $H3H'-teCKOH DJIOCKOCTH. OtteBH.D .HO, IITO npe.11e.n t = 0 B
o6nacTH ro.11orpatt>a Ha puc. 11, 6 .11octnraercsi tor.na, Kor.11a W =
= - 00 B 06.nacTH w Ha puc. 11, 8 HJIH, 'lTO 9KBHBaJieHTHO, Kor.na
T = 0. I -fa 3TOrO CJie.nyeT a = 0 B tt>opMyne (3). c TO'lHOCTblO .110
no.no6HSI Mbl MO>KeM Tenepb H8llHC8Tb paBeHCTBa
c
T=c2_2cc+1•
W=ln T, C= -d/c.
(5)
To11Ka crpyH, ne>Kaw.aH Ha 6ecKoHettHocrn, r.11e W = + oo, otte-
BHJJ.HO, COOTBeTCTByeT 3H8'leHHIO t = eia, (~ + t- 1)/2 =a= c =